neuron-astrocyte-memory-model.txt

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title: "Neuron-Astrocyte Memory Model"
author: "Get Plus"
tags: ["epistemology", "cognition"]
type: "theory"
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Neuron-Astrocyte Memory Model

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Neuron���astrocyte associative memory

Leo Kozachkova,b ID, Jean-Jacques Slotinea,cID, and Dmitry Krotovd,1ID

Af f iliations are included on p. 7.

Edited by Wesley P. Clawson, Allen Discovery Center at Tufts University, Medford, MA; received September 11, 2024; accepted April 7, 2025

by Editorial Board Member Michael S. Gazzaniga

Astrocytes, the most abundant type of glial cell, play a fundamental role in memory.

Despite most hippocampal synapses being contacted by an astrocyte, there are no

current theories that explain how neurons, synapses, and astrocytes might collectively

contributetomemory function.Wedemonstratethat fundamentalaspectsofastrocyte

morphology and physiology naturally lead to a dynamic, high-capacity associative

memory system. The neuron���astrocyte networks generated by our framework are

closely related to popular machine learning architectures known as Dense Associative

Memories. Adjusting the connectivity pattern, the model developed here leads to a

family of associative memory networks that includes a Dense Associative Memory

and a Transformer as two limiting cases. In the known biological implementations of

Dense Associative Memories, the ratio of stored memories to the number of neurons

remains constant, despite the growth of the network size. Our work demonstrates

that neuron���astrocyte networks follow a superior memory scaling law, outperforming

known biological implementations of Dense Associative Memory. Our model suggests

an exciting and previously unnoticed possibility that memories could be stored, at least

in part, within the network of astrocyte processes rather than solely in the synaptic

weights between neurons.

neuron���astrocyte network|associative memory|dynamical

system

Not all brain cells are neurons. It is estimated that about half of the cells in the human

brain are glial cells (from ���glue��� in Greek) (1). Glial cells have long been known to

play an important role in homeostatic brain functions, such as regulating blood l ow

(2)���thus contributing to hemodynamic signals such as those measured in fMRI (3)���

and removing synaptic debris. Converging lines of recent evidence strongly suggest that

they are also directly involved in learning, memory, and cognition (4���10). Among glial

cells, astrocytes are particularly important for brain function. They serve a crucial role in

directly sensing neural activity and, in turn, regulating synaptic strength and plasticity

(4, 5, 11���14). In addition to sensing neural activity, astrocytes are also important targets

of neuromodulatory signals such as norepinephrine and acetylcholine emerging from

potentially distant brain structures such as the brainstem (15).

Of particular relevance to the computational neuroscience community are the recent

i ndings that 1) astrocytes are necessary for forming and retrieving long-term memories

(i.e., by participating in engrams) (6, 16���20) and 2) astrocytes respond to neural activity

on timescales spanning many orders of magnitude, from several hundred milliseconds to

minutes (14, 21, 22). Despite extensive evidence establishing the importance of neuron���

astrocyte interactions for long-term memory function, computational theories of these

interactions are still in their infancy.

1.1. What Shapes Astrocytic Computation?The core proposal of this paper is that

astrocytes compute, and these computations are shaped by tunable signaling pathways

within astrocytes. We will be primarily concerned with associative computations: How

neurons, synapses, and astrocytes work together to store and retrieve memories. In

this case, astrocytic Ca2+f l ux coeff i cients are the site of memory storage, and neuron���

synapse���astrocyte interactions are the mechanism of memory retrieval. This proposal

harmoniously extends decades of prior work suggesting that memories are stored in

synaptic weights (23, 24) and provides a perspective where synaptic weights ���emerge���

from interactions between neurons and astrocytes. Note that our proposal remains

consistent with the engram hypothesis, in emphasizing that neural activation during

a learning event is essential for subsequent memory recall (25, 26). Indeed, our work

closely aligns with recent experimental i ndings indicating that astrocytes collaborate

with neurons to store and retrieve memories through engram representations (20).

Signif i cance

Recent experiments have

challenged the belief that glial

cells, which compose at least half

of brain cells, are just passive

support structures. Despite this,

a clear understanding of how

neurons and glia work together

for brain function is missing. To

close this gap, we present a

theory of neuron���astrocytes

networks for memory processing,

using the Dense Associative

Memory framework. Our findings

suggest that astrocytes can serve

as natural units for implementing

this network in biological

���hardware.��� Astrocytes enhance

the memory capacity of the

network. This boost originates

from storing memories in the

network of astrocytic processes,

not just in synapses, as

commonly believed. These

process-to-process

communications likely occur in

the brain and could help explain

its impressive memory

processing capabilities.

Author contributions: L.K. and D.K. designed research;

L.K., J.-J.S., and D.K. performed research; and L.K., J.-J.S.,

and D.K. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission. W.P.C. is a guest

editor invited by the Editorial Board.

Copyright��2025 the Author(s). Published by PNAS.

This open access article is distributed under Creative

Commons Attribution License 4.0 (CC BY).

1To whom correspondence may be addressed. Email:

krotov@ibm.com.

This article contains supporting information online

athttps://www.pnas.org/lookup/suppl/doi:10.1073/pnas.

2417788122/-/DCSupplemental.

Published May 23, 2025.

PNAS2025Vol. 122No. 21e2417788122https://doi.org/10.1073/pnas.24177881221 of 7

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2. Neuron���Astrocyte Model

Astrocytes have a primary cell body (soma) with numerous

branching processes that envelope nearby synapses (Fig. 1). This

three-part structure is known as the tripartite synapse (27). A

single astrocyte can form over 106tripartite synapses (28), and

astrocyte networks spatially tile the brain, forming nonoverlap-

ping ���islands��� (29). Astrocyte processes detect neurotransmitters

in the synaptic cleft, leading to an upsurge in intracellular free

calcium Ca2+ions within the astrocyte process. This leads to

a biochemical cascade in the astrocyte, potentially culminating

in the release of gliotransmitters back into the synaptic cleft,

inf l uencing neural activity���a closed feedback loop. Astrocyte

processes can intercommunicate through calcium transport (30),

andindividualastrocytesconnectviagapjunctions.Theinterplay

betweenneuronsandastrocytes,spanningmultipletemporaland

spatial scales, underscores the relevance of astrocytes in learning

andmemory.Forthispaper,wewillfocusonthefollowingsalient

aspects of astrocyte biology:

��� A single astrocyte can connect to millions of nearby synapses,

forming three-part connections (astrocyte process, presynaptic

neuron, postsynaptic neuron) called tripartite synapses (27).

��� Astrocytes detect neural activity and respond by regulating this

activity through the release of gliotransmitters (31).

��� Tripartite synapses can interact with each other, possibly

through astrocytic intracellular calcium transport (30).

2.1. Neural Dynamics.The above points may be formalized into

a set of dynamical equations governing the time evolution of

neurons, synapses, and astrocytes. The membrane voltage xifor

each neuron i evolves according to a standard rate recurrent

neural network model (32, 33) with the characteristic time scale

of the neural dynamics ���n, and the leak rate ���

���n�� xi= ������ xi+

N X

j=1

g(sij) ���(xj) + bi[1]

Each neuron has an input bi, which establishes the neuron���s

baseline activation. The nonlinearity ���(xj) transforms neural

membranevoltagesintof i ringrates,whilethetermg(sij)indicates

the strength of the synaptic weight connecting neurons i and j.

Thevariablesijisdynamicandalters dependingontheactivityof

AB

Fig. 1.(A) An abstracted version of an astrocyte, showing the astrocyte

processes and the synapses. (B) Our mathematical idealization of the

minicircuit defined by a single astrocyte.

Box 1.

ExamplesofpossibleLagrangianfunctions.

Here, the variable z is an arbitrary dynamical

variable in our model (e.g., astrocyte calcium

level). Recall from the main text that activation

functions are def i ned from the Lagrangians as

���L

���zi. The i rst Lagrangian provides an example

of a ���collective��� activation function. The second

Lagrangian leads to an element-wise activa-

tion function, assuming ���Q

���zi = q(zi). Generally,

the only mathematical requirement for our La-

grangians is that they must be convex functions.

Examples of Possible Lagrangians and Activations

L(z) = log

N X

i=1 ezi��� ���L(z)

���z = Softmax(z)

L(z) =

N X

i=1 Q(zi)��� ���L(z)

���z

ChatGPT said:

This paper���

���Neuron���astrocyte associative memory��� by Kozachkov, Slotine, and Krotov (2025,

PNAS

)

���presents a groundbreaking computational model that places

astrocytes

, long relegated to passive support roles in the brain, at the center of

dynamic, high-capacity associative memory

. Here���s a clear breakdown and key implications:

����

Core Ideas

1.

Astrocytes as Active Memory Units

Traditionally

: Memory is thought to be stored in

synaptic weights

between neurons.

In this model

: Astrocytes���through

Ca����� flux dynamics

���play a central role in

storing

and

modulating

memory. The astrocytic processes interacting with tripartite synapses are not just passive enablers, but

computational components

.

2.

Tripartite Synapse as a Computational Unit

Each memory "bit" involves:

A

presynaptic neuron

A

postsynaptic neuron

An

astrocytic process

enveloping the synapse

This forms a

tripartite unit

, where astrocytic Ca����� activity responds to neurotransmitter release, integrates information, and can

modulate synaptic strength

via gliotransmitters.

Astrocytes intercommunicate via

intracellular calcium waves

, suggesting

spatially distributed

memory processing.

3.

Mathematical Framework: Dense Associative Memories (DAMs)

The authors develop a neuron���astrocyte system that generalizes the

Dense Associative Memory

(DAM) architecture���a known high-capacity memory model.

Key features:

DAMs can store and retrieve

patterns

robustly by

energy minimization

in a dynamical system.

The model extends to

Transformer-like architectures

depending on the connectivity���suggesting a

unified memory-computation landscape

spanning biological and machine intelligence.

4.

Superior Scaling Laws

Biological DAM implementations scale linearly

, meaning that memory capacity grows proportionally with the number of neurons.

Neuron���astrocyte networks scale super-linearly

, implying

greater-than-expected memory capacity

as the network grows.

This is because

astrocytic processes can form millions of synaptic interactions

, creating

combinatorially rich, non-local memory associations

beyond what synapse-only models permit.

����

Model Components

���� Neuronal Dynamics:

Rate-based neural model:

��

n

x

��

i

=

���

��

x

i

+

���

j

=

1

N

g

(

s

i

j

)

��

(

x

j

)

+

b

i

\tau_n \dot{x}_i = -\lambda x_i + \sum_{j=1}^N g(s_{ij}) \phi(x_j) + b_i

��

n

���

x

��

i

���

=

���

��

x

i

���

+

j

=

1

���

N

���

g

(

s

ij

���

)

��

(

x

j

���

)

+

b

i

���

x

i

x_i

x

i

���

: membrane voltage of neuron

i

i

i

s

i

j

s_{ij}

s

ij

���

: synaptic weight (dynamic, activity-dependent)

��

\phi

��

: activation function

b

i

b_i

b

i

���

: baseline input

���� Astrocytic Dynamics:

Astrocyte processes:

Detect synaptic neurotransmitter release

Increase internal

Ca

2

+

\text{Ca}^{2+}

Ca

2

+

Release gliotransmitters back to neurons

Communicate with each other through calcium waves and gap junctions

These dynamics are encoded using

Lagrangian-based activation functions

for generality, including:

Softmax

from log-sum-exp Lagrangians

Element-wise activations

from separable convex potentials

����

Implications

1.

Memories May Be Stored in Astrocytic Networks

Not just synaptic weights, but the

calcium state-space

and

inter-process dynamics

of astrocytes may

encode long-term memory patterns

.

2.

Biology as High-Capacity Associative Hardware

Astrocytes appear optimized to implement architectures similar to those in

deep learning

:

Massive parallelism

Dynamical recurrence

Gradient-like integration via biochemical signaling

3.

Synaptic Weights May Be Emergent

Rather than being the

primary

locus of memory, synaptic weights may emerge from

deeper neuron���astrocyte dynamics

���flipping the classical paradigm.

���� Takeaways

Concept

Traditional View

This Paper

Memory Storage

Synaptic weights

Astrocytic Ca����� + synapse interactions

Memory Scaling

Linear with neuron count

Super-linear via astrocyte processes

Astrocyte Role

Support

Active computation and storage

Model Type

Recurrent neural net / DAM

DAM + Transformer generalization with Lagrangian activations

If you'd like, I can help you:

Diagram the neuron���astrocyte interaction model

Derive the full dynamical system from the equations

Explore implications for AI memory architectures

Compare with engram theory or Predictive Processing models

Would you like to go deeper in one of those directions?

You said:

2. Neuron���Astrocyte Model

Astrocytes have a primary cell body (soma) with numerous

branching processes that envelope nearby synapses (Fig. 1). This

three-part structure is known as the tripartite synapse (27). A

single astrocyte can form over 106tripartite synapses (28), and

astrocyte networks spatially tile the brain, forming nonoverlap-

ping ���islands��� (29). Astrocyte processes detect neurotransmitters

in the synaptic cleft, leading to an upsurge in intracellular free

calcium Ca2+ions within the astrocyte process. This leads to

a biochemical cascade in the astrocyte, potentially culminating

in the release of gliotransmitters back into the synaptic cleft,

inf l uencing neural activity���a closed feedback loop. Astrocyte

processes can intercommunicate through calcium transport (30),

andindividualastrocytesconnectviagapjunctions.Theinterplay

betweenneuronsandastrocytes,spanningmultipletemporaland

spatial scales, underscores the relevance of astrocytes in learning

andmemory.Forthispaper,wewillfocusonthefollowingsalient

aspects of astrocyte biology:

��� A single astrocyte can connect to millions of nearby synapses,

forming three-part connections (astrocyte process, presynaptic

neuron, postsynaptic neuron) called tripartite synapses (27).

��� Astrocytes detect neural activity and respond by regulating this

activity through the release of gliotransmitters (31).

��� Tripartite synapses can interact with each other, possibly

through astrocytic intracellular calcium transport (30).

2.1. Neural Dynamics.The above points may be formalized into

a set of dynamical equations governing the time evolution of

neurons, synapses, and astrocytes. The membrane voltage xifor

each neuron i evolves according to a standard rate recurrent

neural network model (32, 33) with the characteristic time scale

of the neural dynamics ���n, and the leak rate ���

���n�� xi= ������ xi+

N X

j=1

g(sij) ���(xj) + bi[1]

Each neuron has an input bi, which establishes the neuron���s

baseline activation. The nonlinearity ���(xj) transforms neural

membranevoltagesintof i ringrates,whilethetermg(sij)indicates

the strength of the synaptic weight connecting neurons i and j.

Thevariablesijisdynamicandalters dependingontheactivityof

AB

Fig. 1.(A) An abstracted version of an astrocyte, showing the astrocyte

processes and the synapses. (B) Our mathematical idealization of the

minicircuit defined by a single astrocyte.

Box 1.

ExamplesofpossibleLagrangianfunctions.

Here, the variable z is an arbitrary dynamical

variable in our model (e.g., astrocyte calcium

level). Recall from the main text that activation

functions are def i ned from the Lagrangians as

���L

���zi. The i rst Lagrangian provides an example

of a ���collective��� activation function. The second

Lagrangian leads to an element-wise activa-

tion function, assuming ���Q

���zi = q(zi). Generally,

the only mathematical requirement for our La-

grangians is that they must be convex functions.

Examples of Possible Lagrangians and Activations

L(z) = log

N X

i=1 ezi��� ���L(z)

���z = Softmax(z)

L(z) =

N X

i=1 Q(zi)��� ���L(z)

���z

= ?q(z

1),...,q(zn) ?> [2]

both neurons and astrocytes, as we will detail next. Note that for

i xedsij,thismodelissimplyastandardrecurrentneuralnetwork.

2.2. Synaptic Dynamics.The level of synaptic facilitation, which

isdenotedbysij,referstothedegreetowhichpresynapticspiking

activity impacts the postsynaptic neuron. Biophysically, there are

manyfactorswhichinf l uencetheeff i cacyofasynapse,suchasthe

number of postsynaptic receptors and the level of calcium in the

synaptic cleft. The strength of the synapse can either increase or

decreasebasedonbothpre-andpostsynapticactivity,asobserved

in Hebbian plasticity. As in earlier studies of neuron���glial

interactions, we consider tripartite synapses���synapses whose

plasticity is modulated by an enveloping astrocytic process, pij

���s�� sij= ������ sij+ f (xi,xj,pij,sij) + cij[3]

The timescale of the synaptic dynamics is ���s, and ��� establishes

the leak-rate of synaptic facilitation. The function f encapsulates

the interactions between these three biological variables. The

inputs cijserve as bias variables, controlling the baseline rate

of synaptic facilitation in the absence of external input. The

concentration of intracellular Ca2+ions in the astrocytic process

that wraps around the particular synapse i ��� j is denoted by pij.

Biophysically, astrocytes inf l uence neural activity through Ca2+-

dependentexocytosisofgliotransmitterssuchasGABA,D-serine,

ATP, and glutamate (14)���an inf l uence which is encoded in the

function f . The level of intracellular astrocytic Ca2+is a dynamic

variable and its value depends on both the calcium levels in

adjacent astrocyte processes and the synaptic state sij.

2.3. Astrocyte Process Dynamics.The state of a specif i c astro-

cytic process is determined by its interactions with neurons at

the tripartite synapse, plus its interactions with other processes

through intracellular calcium transport

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���p�� pij= ������ pij+

N X

k,l=1

Tijkl���(pkl) + ���(sij) + dij[4]

The double sum in the astrocyte equations captures the interac-

tions between process pijand all other processes (34). In the sim-

plest scenario, calcium can diffuse between processes, resulting

in a linear function ��� and tensor Tijkldescribing concentration

l uxes. More complex calcium transport mechanisms between

different processes within an astrocyte can produce nonlinear

functions ���. The term dijis a constant bias term which sets the

overall���tone���oftheastrocyte.Thisvariablemaybethoughtofas

a neuromodulatory signal, potentially arriving from distant brain

regions such as the pons. The input from process pklto process

pijis weighed by the scalar Tijkl. A zero value for this scalar

indicates no direct physical connection between process ij and

kl. Hence, the astrocyte���s anatomical structure can be encoded

in the nonzero entries of tensor T. The nonlinear function

��� encapsulates the synapse ��� astrocyte signaling pathway at

the tripartite synapse level. Here, ���prepresents the astrocyte

timescale, while ��� > 0 is a leak term for the intracellular calcium

in the astrocyte process.

3. Associative Neuron���Astrocyte Model

In Section 2, we described a general framework, grounded in

the biology of neuron���astrocyte communication, for modeling

neuron���astrocyte interactions via the tripartite synapses. De-

pending on the choices of the nonlinearities and parameters this

network can exhibit many sophisticated dynamical behaviors���

such as chaos or limit cycles���which can be diff i cult to analyze

in full generality.

To better understand the potential role of neuron���astrocyte

interactions, we will focus on an important limiting case where

the system demonstrates associative memory functions. As in

essentially all models of biological associative memory, this

requiresthepresenceofsymmetriesinthegoverningequationsof

thebiologicalcircuit.Weshowthatundercertainconditions,the

resulting neuron���astrocyte model has a global energy function

(Lyapunov function), which monotonically decreases on the

dynamical trajectory and is bounded from below. This makes it

possibletoidentifyaregimeofoperationoftheneuron���astrocyte

network that results in dynamical trajectories converging to

i xed point attractor states. The i xed points can be identif i ed

with ���memories��� stored in the weight matrices, and the entire

neuron���astrocyte model can be regarded as an energy-based

Dense Associative Memory (also known as Modern Hopf i eld

Network) (35, 36). Importantly, this framework allows us to

show that the presence of a single astrocyte can provably boost

the memory capacity per compute unit of a neural circuit by a

factor of N.

We will follow the general formulation of energy-based

associative memories (37���39), which starts with picking three

Lagrangians, which def i ne layers of our architecture (neurons,

synapses, and astrocytic processes), and the corresponding acti-

vationfunctions.TheseLagrangiansareaneuralLagrangianL[n],

a synaptic Lagrangian L[s], and an astrocyte process Lagrangian

L[p]. In general these scalar functions can be arbitrary (differen-

tiable) functions of the corresponding dynamical variables. The

details of our derivation can be found in SI Appendix, section 1.

From these Lagrangians, we can derive via a Legendre

transformation three terms in the overall energy function of

the neuron���astrocyte system: E[n], E[s], and E[p]. The activation

functions in our model are dictated by the Lagrangians���indeed,

theithactivationissimplythepartialderivativeoftheLagrangian

with respect to the ithdynamical variable (SI Appendix, section 1

andBox1).Theremainingcontributionstothetotalenergyofthe

system describe the interactions between neurons, synapses, and

astrocytes. These contributions describe the synapse-mediated

interactions between the neurons E[ns], the interactions between

theprocessesandthesynapsesE[ps],andtheinteractionsbetween

the individual processes inside the astrocyte E[pp].

The overall energy function of the neuron���astrocyte model

can now be written as the sum of these six terms

E = E[n]+ E[s]+ E[p]+ E[ns]+ E[ps]+ E[pp][5]

From this Lagrangian formalism (37���39), the dynamical equa-

tions for the associative neuron���astrocyte energy can be viewed

as the negative gradient (with respect to the nonlinearities):

��� ��� ���

ChatGPT said:

You're diving into a remarkable piece of work. The

Neuron���Astrocyte Associative Memory

model by Kozachkov, Slotine, and Krotov presents a

novel tripartite dynamical system

uniting

neurons

,

synapses

, and

astrocytes

into a

multi-layer energy-based memory network

. Let���s unpack Section 2 and Section 3 of the paper���focusing on the biological grounding, dynamical equations, and how it transitions into an

energy-minimizing associative memory system

.

���� SECTION 2: Neuron���Astrocyte Model

This section formalizes the

three interlinked dynamical components

of the system:

���� 2.1 Neuronal Dynamics

Each neuron's membrane voltage

x

i

x_i

x

i

���

evolves according to a

leaky rate-based model

:

��

n

x

��

i

=

���

��

x

i

+

���

j

=

1

N

g

(

s

i

j

)

��

(

x

j

)

+

b

i

(1)

\tau_n \dot{x}_i = -\lambda x_i + \sum_{j=1}^N g(s_{ij}) \phi(x_j) + b_i \tag{1}

��

n

���

x

��

i

���

=

���

��

x

i

���

+

j

=

1

���

N

���

g

(

s

ij

���

)

��

(

x

j

���

)

+

b

i

���

(

1

)

��

(

x

j

)

\phi(x_j)

��

(

x

j

���

)

: nonlinear activation (e.g. tanh, ReLU)

g

(

s

i

j

)

g(s_{ij})

g

(

s

ij

���

)

: function of synaptic strength (modulated by astrocyte interaction)

b

i

b_i

b

i

���

: bias input

If

s

i

j

s_{ij}

s

ij

���

were static, this is a vanilla recurrent neural network.

���� 2.2 Synaptic Dynamics

The synapse strength

s

i

j

s_{ij}

s

ij

���

between neurons

i

i

i

and

j

j

j

is

plastic

and

modulated by astrocytic calcium levels

p

i

j

p_{ij}

p

ij

���

:

��

s

s

��

i

j

=

���

��

s

i

j

+

f

(

x

i

,

x

j

,

p

i

j

,

s

i

j

)

+

c

i

j

(2)

\tau_s \dot{s}_{ij} = -\alpha s_{ij} + f(x_i, x_j, p_{ij}, s_{ij}) + c_{ij} \tag{2}

��

s

���

s

��

ij

���

=

���

��

s

ij

���

+

f

(

x

i

���

,

x

j

���

,

p

ij

���

,

s

ij

���

)

+

c

ij

���

(

2

)

f

f

f

: activity-dependent Hebbian-like function including astrocyte influence

c

i

j

c_{ij}

c

ij

���

: synaptic bias

p

i

j

p_{ij}

p

ij

���

: astrocyte process calcium level surrounding synapse

i

���

j

i-j

i

���

j

Here,

astrocytic calcium levels directly participate in synaptic learning rules

.

���� 2.3 Astrocytic Process Dynamics

Each

astrocytic process

around a synapse evolves via calcium exchange and feedback from neuron-synapse activity:

��

p

p

��

i

j

=

���

��

p

i

j

+

���

k

,

l

=

1

N

T

i

j

k

l

��

(

p

k

l

)

+

��

(

s

i

j

)

+

d

i

j

(3)

\tau_p \dot{p}_{ij} = -\gamma p_{ij} + \sum_{k,l=1}^N T_{ijkl} \psi(p_{kl}) + \kappa(s_{ij}) + d_{ij} \tag{3}

��

p

���

p

��

���

ij

���

=

���

��

p

ij

���

+

k

,

l

=

1

���

N

���

T

ijk

l

���

��

(

p

k

l

���

)

+

��

(

s

ij

���

)

+

d

ij

���

(

3

)

��

(

p

k

l

)

\psi(p_{kl})

��

(

p

k

l

���

)

: nonlinear interaction (e.g. diffusion) from other astrocyte processes

��

(

s

i

j

)

\kappa(s_{ij})

��

(

s

ij

���

)

: calcium influx from synaptic state

T

i

j

k

l

T_{ijkl}

T

ijk

l

���

: structural tensor defining which processes are interconnected

d

i

j

d_{ij}

d

ij

���

: neuromodulatory tone (e.g. acetylcholine from pons)

This formalism supports

distributed computation

over calcium fields inside astrocyte trees.

���� SECTION 3: Associative Neuron���Astrocyte Memory

To give the system

associative memory properties

, the authors

construct an energy function

from the three layers using

Lagrangian dynamics

.

���� Lagrangian-Driven Structure

Each of the three variables���neurons

x

i

x_i

x

i

���

, synapses

s

i

j

s_{ij}

s

ij

���

, and astrocyte calcium

p

i

j

p_{ij}

p

ij

���

���has a

Lagrangian

:

L

[

n

]

(

x

)

\mathcal{L}^{[n]}(x)

L

[

n

]

(

x

)

��� neuron layer

L

[

s

]

(

s

)

\mathcal{L}^{[s]}(s)

L

[

s

]

(

s

)

��� synapse layer

L

[

p

]

(

p

)

\mathcal{L}^{[p]}(p)

L

[

p

]

(

p

)

��� astrocyte layer

Activation functions are derived from gradients of Lagrangians:

Activation:

���

L

���

z

i

���

��

(

x

i

)

,

g

(

s

i

j

)

,

��

(

p

i

j

)

\text{Activation: } \frac{\partial \mathcal{L}}{\partial z_i} \Rightarrow \phi(x_i), g(s_{ij}), \psi(p_{ij})

Activation:

���

z

i

���

���

L

���

���

��

(

x

i

���

)

,

g

(

s

ij

���

)

,

��

(

p

ij

���

)

Example from Box 1:

Softmax from log-sum-exp:

L

(

z

)

=

log

���

���

i

e

z

i

���

���

L

���

z

i

=

Softmax

(

z

)

i

\mathcal{L}(z) = \log \sum_i e^{z_i} \Rightarrow \frac{\partial \mathcal{L}}{\partial z_i} = \text{Softmax}(z)_i

L

(

z

)

=

lo

g

i

���

���

e

z

i

���

���

���

z

i

���

���

L

���

=

Softmax

(

z

)

i

���

���� Full Energy Function

The total system energy is:

E

=

E

[

n

]

+

E

[

s

]

+

E

[

p

]

+

E

[

n

s

]

+

E

[

p

s

]

+

E

[

p

p

]

(4)

E = E^{[n]} + E^{[s]} + E^{[p]} + E^{[ns]} + E^{[ps]} + E^{[pp]} \tag{4}

E

=

E

[

n

]

+

E

[

s

]

+

E

[

p

]

+

E

[

n

s

]

+

E

[

p

s

]

+

E

[

pp

]

(

4

)

Where:

E

[

n

]

,

E

[

s

]

,

E

[

p

]

E^{[n]}, E^{[s]}, E^{[p]}

E

[

n

]

,

E

[

s

]

,

E

[

p

]

: internal energy of neurons, synapses, astrocyte processes

E

[

n

s

]

E^{[ns]}

E

[

n

s

]

: neuron-synapse interactions

E

[

p

s

]

E^{[ps]}

E

[

p

s

]

: astrocyte-synapse coupling

E

[

p

p

]

E^{[pp]}

E

[

pp

]

: astrocyte���astrocyte interactions via calcium waves

Under symmetric connectivity conditions,

this energy decreases over time

, yielding a

Lyapunov function

. The system thus

settles into stable attractors

representing stored memories.

���� What This Means

1.

Astrocytes Boost Associative Capacity

A single astrocyte interacting with

N

N

N

synapses can increase memory capacity by

a factor of

N

N

N

compared to neuron-only models.

This is due to

combinatorial richness

in astrocyte���synapse interactions and distributed calcium signaling.

2.

Energy-Based Memory with Biological Plausibility

The model is a

biological Dense Associative Memory (DAM)

, similar in spirit to a

Hopfield network

, but grounded in real cellular dynamics.

3.

Astrocyte Networks Mirror Transformer-Like Behavior

By adjusting connectivity and activation Lagrangians, the model interpolates between:

Dense Associative Memory

(DAM) behavior

Transformer-like architectures

(attention + nonlinear aggregation)

This suggests a

deep homology

between biological memory and modern deep learning.

���� TL;DR Summary

Component

Variable

Equation

Description

Neuron

x

i

x_i

x

i

���

(1)

RNN dynamics w/ synaptic input

Synapse

s

i

j

s_{ij}

s

ij

���

(2)

Hebbian + astrocyte-modulated plasticity

Astrocyte Process

p

i

j

p_{ij}

p

ij

���

(3)

Calcium-based interprocess and synapse feedback

System Energy

E

E

E

(4)

Lyapunov function for memory attractors

Memory Boost

���

���

Memory scales superlinearly via astrocytic structure

Would you like help visualizing this with a

network diagram

or want a

step-by-step derivation of the energy gradient flow

from the Lagrangians?

You said:

A step by step derivation

ChatGPT said:

Great ��� let���s walk through a

step-by-step derivation

of the

neuron���astrocyte associative memory system

based on the

Lagrangian formalism

presented in the paper.

This framework constructs an

energy function

E

E

E

for the system, using

three interacting layers

��� neurons

{

x

i

}

\{x_i\}

{

x

i

���

}

, synapses

{

s

i

j

}

\{s_{ij}\}

{

s

ij

���

}

, and astrocytic processes

{

p

i

j

}

\{p_{ij}\}

{

p

ij

���

}

. The

activation dynamics

of each layer are governed by the

gradient of a convex Lagrangian

with respect to its variables.

We will derive:

The

layer-specific energy contributions

The

interaction terms

The

overall dynamics as gradient descent on energy

������ STEP 1: Define Layer Variables

Let:

x

���

R

N

\mathbf{x} \in \mathbb{R}^N

x

���

R

N

: vector of neuron membrane voltages

S

���

R

N

��

N

\mathbf{S} \in \mathbb{R}^{N \times N}

S

���

R

N

��

N

: synaptic facilitation matrix

P

���

R

N

��

N

\mathbf{P} \in \mathbb{R}^{N \times N}

P

���

R

N

��

N

: astrocyte process calcium levels (one per tripartite synapse)

Each variable will have:

A

Lagrangian

:

L

[

x

]

(

x

)

\mathcal{L}^{[x]}(\mathbf{x})

L

[

x

]

(

x

)

,

L

[

s

]

(

S

)

\mathcal{L}^{[s]}(\mathbf{S})

L

[

s

]

(

S

)

,

L

[

p

]

(

P

)

\mathcal{L}^{[p]}(\mathbf{P})

L

[

p

]

(

P

)

A

corresponding energy

:

E

[

x

]

,

E

[

s

]

,

E

[

p

]

E^{[x]}, E^{[s]}, E^{[p]}

E

[

x

]

,

E

[

s

]

,

E

[

p

]

���� STEP 2: Layer Lagrangians and Energy Contributions

Each Lagrangian encodes a

prior structure

or

activation shape

:

Example 1: Softmax-like Lagrangian for neurons

L

[

x

]

(

x

)

=

log

���

(

���

i

=

1

N

e

x

i

)

���

���

L

[

x

]

���

x

i

=

softmax

(

x

)

i

\mathcal{L}^{[x]}(\mathbf{x}) = \log \left( \sum_{i=1}^N e^{x_i} \right) \quad \Rightarrow \quad \frac{\partial \mathcal{L}^{[x]}}{\partial x_i} = \text{softmax}(x)_i

L

[

x

]

(

x

)

=

lo

g

(

i

=

1

���

N

���

e

x

i

���

)

���

���

x

i

���

���

L

[

x

]

���

=

softmax

(

x

)

i

���

Example 2: Separable elementwise Lagrangian

L

[

x

]

(

x

)

=

���

i

Q

(

x

i

)

���

���

L

[

x

]

���

x

i

=

q

(

x

i

)

\mathcal{L}^{[x]}(\mathbf{x}) = \sum_i Q(x_i) \quad \Rightarrow \quad \frac{\partial \mathcal{L}^{[x]}}{\partial x_i} = q(x_i)

L

[

x

]

(

x

)

=

i

���

���

Q

(

x

i

���

)

���

���

x

i

���

���

L

[

x

]

���

=

q

(

x

i

���

)

This gives

activation functions

directly:

��

(

x

i

)

=

���

L

[

x

]

���

x

i

\phi(x_i) = \frac{\partial \mathcal{L}^{[x]}}{\partial x_i}

��

(

x

i

���

)

=

���

x

i

���

���

L

[

x

]

���

g

(

s

i

j

)

=

���

L

[

s

]

���

s

i

j

g(s_{ij}) = \frac{\partial \mathcal{L}^{[s]}}{\partial s_{ij}}

g

(

s

ij

���

)

=

���

s

ij

���

���

L

[

s

]

���

��

(

p

i

j

)

=

���

L

[

p

]

���

p

i

j

\psi(p_{ij}) = \frac{\partial \mathcal{L}^{[p]}}{\partial p_{ij}}

��

(

p

ij

���

)

=

���

p

ij

���

���

L

[

p

]

���

The energy contribution of each layer is the

Legendre transform

:

E

[

x

]

=

���

i

(

x

i

���

��

(

x

i

)

���

L

[

x

]

(

x

i

)

)

E^{[x]} = \sum_i \left( x_i \cdot \phi(x_i) - \mathcal{L}^{[x]}(x_i) \right)

E

[

x

]

=

i

���

���

(

x

i

���

���

��

(

x

i

���

)

���

L

[

x

]

(

x

i

���

)

)

(similarly for

E

[

s

]

E^{[s]}

E

[

s

]

and

E

[

p

]

E^{[p]}

E

[

p

]

).

���� STEP 3: Define Interaction Terms

Now define

coupling energies

between layers:

1. Neuron���Synapse interaction:

E

[

x

s

]

=

���

���

i

,

j

g

(

s

i

j

)

���

��

(

x

j

)

���

x

i

E^{[xs]} = -\sum_{i,j} g(s_{ij}) \cdot \phi(x_j) \cdot x_i

E

[

x

s

]

=

���

i

,

j

���

���

g

(

s

ij

���

)

���

��

(

x

j

���

)

���

x

i

���

This reflects

postsynaptic influence

of neuron

j

j

j

on neuron

i

i

i

weighted by synaptic strength.

2. Synapse���Astrocyte interaction:

E

[

s

p

]

=

���

���

i

,

j

��

(

s

i

j

)

���

p

i

j

E^{[sp]} = -\sum_{i,j} \kappa(s_{ij}) \cdot p_{ij}

E

[

s

p

]

=

���

i

,

j

���

���

��

(

s

ij

���

)

���

p

ij

���

Where

��

(

s

)

\kappa(s)

��

(

s

)

captures how astrocyte processes are influenced by synapse state (e.g., Ca����� influx from synaptic activity).

3. Astrocyte���Astrocyte interaction:

E

[

p

p

]

=

���

i

,

j

,

k

,

l

T

i

j

k

l

���

��

(

p

k

l

)

���

p

i

j

E^{[pp]} = \sum_{i,j,k,l} T_{ijkl} \cdot \psi(p_{kl}) \cdot p_{ij}

E

[

pp

]

=

i

,

j

,

k

,

l

���

���

T

ijk

l

���

���

��

(

p

k

l

���

)

���

p

ij

���

This is the

calcium wave propagation

within astrocyte trees ��� a quadratic-like energy from diffusion or gap junctions.

���� STEP 4: Total Energy Function

Add up all components:

E

[

x

,

S

,

P

]

=

E

[

x

]

+

E

[

s

]

+

E

[

p

]

+

E

[

x

s

]

+

E

[

s

p

]

+

E

[

p

p

]

(5)

E[\mathbf{x}, \mathbf{S}, \mathbf{P}] = E^{[x]} + E^{[s]} + E^{[p]} + E^{[xs]} + E^{[sp]} + E^{[pp]} \tag{5}

E

[

x

,

S

,

P

]

=

E

[

x

]

+

E

[

s

]

+

E

[

p

]

+

E

[

x

s

]

+

E

[

s

p

]

+

E

[

pp

]

(

5

)

This total energy is

Lyapunov

: it

monotonically decreases over time

under the dynamics defined below.

���� STEP 5: Gradient Flow Dynamics

For each dynamical variable

z

���

{

x

i

,

s

i

j

,

p

i

j

}

z \in \{ x_i, s_{ij}, p_{ij} \}

z

���

{

x

i

���

,

s

ij

���

,

p

ij

���

}

, the dynamics follow

gradient descent

on

E

E

E

:

��

z

���

z

��

=

���

���

E

���

z

\tau_z \cdot \dot{z} = - \frac{\partial E}{\partial z}

��

z

���

���

z

��

=

���

���

z

���

E

���

So:

Neuron:

��

x

���

x

��

i

=

���

���

E

���

x

i

=

���

(

���

E

[

x

]

���

x

i

+

���

E

[

x

s

]

���

x

i

)

\tau_x \cdot \dot{x}_i = - \frac{\partial E}{\partial x_i} = - \left( \frac{\partial E^{[x]}}{\partial x_i} + \frac{\partial E^{[xs]}}{\partial x_i} \right)

��

x

���

���

x

��

i

���

=

���

���

x

i

���

���

E

���

=

���

(

���

x

i

���

���

E

[

x

]

���

+

���

x

i

���

���

E

[

x

s

]

���

)

Synapse:

��

s

���

s

��

i

j

=

���

(

���

E

[

s

]

���

s

i

j

+

���

E

[

x

s

]

���

s

i

j

+

���

E

[

s

p

]

���

s

i

j

)

\tau_s \cdot \dot{s}_{ij} = - \left( \frac{\partial E^{[s]}}{\partial s_{ij}} + \frac{\partial E^{[xs]}}{\partial s_{ij}} + \frac{\partial E^{[sp]}}{\partial s_{ij}} \right)

��

s

���

���

s

��

ij

���

=

���

(

���

s

ij

���

���

E

[

s

]

���

+

���

s

ij

���

���

E

[

x

s

]

���

+

���

s

ij

���

���

E

[

s

p

]

���

)

Astrocyte process:

��

p

���

p

��

i

j

=

���

(

���

E

[

p

]

���

p

i

j

+

���

E

[

s

p

]

���

p

i

j

+

���

E

[

p

p

]

���

p

i

j

)

\tau_p \cdot \dot{p}_{ij} = - \left( \frac{\partial E^{[p]}}{\partial p_{ij}} + \frac{\partial E^{[sp]}}{\partial p_{ij}} + \frac{\partial E^{[pp]}}{\partial p_{ij}} \right)

��

p

���

���

p

��

���

ij

���

=

���

(

���

p

ij

���

���

E

[

p

]

���

+

���

p

ij

���

���

E

[

s

p

]

���

+

���

p

ij

���

���

E

[

pp

]

���

)

This gives a

gradient descent system

with attractors.

��� STEP 6: Fixed Point = Memory Attractor

A memory is retrieved when the system converges to a

fixed point

:

d

x

d

t

=

0

,

d

S

d

t

=

0

,

d

P

d

t

=

0

\frac{d\mathbf{x}}{dt} = 0, \quad \frac{d\mathbf{S}}{dt} = 0, \quad \frac{d\mathbf{P}}{dt} = 0

d

t

d

x

���

=

0

,

d

t

d

S

���

=

0

,

d

t

d

P

���

=

0

At this point:

The

energy is minimized

The network has

recalled a stored memory pattern

That memory may be stored in

astrocytic state

,

synaptic configuration

, or both

���� Summary of Derivation

Layer

Variable

Lagrangian

Activation

Dynamics

Neuron

x

i

x_i

x

i

���

L

[

x

]

\mathcal{L}^{[x]}

L

[

x

]

��

(

x

i

)

=

���

L

[

x

]

/

���

x

i

\phi(x_i) = \partial \mathcal{L}^{[x]}/\partial x_i

��

(

x

i

���

)

=

���

L

[

x

]

/

���

x

i

���

x

��

i

=

���

���

E

/

���

x

i

\dot{x}_i = -\partial E/\partial x_i

x

��

i

���

=

���

���

E

/

���

x

i

���

Synapse

s

i

j

s_{ij}

s

ij

���

L

[

s

]

\mathcal{L}^{[s]}

L

[

s

]

g

(

s

i

j

)

=

���

L

[

s

]

/

���

s

i

j

g(s_{ij}) = \partial \mathcal{L}^{[s]}/\partial s_{ij}

g

(

s

ij

���

)

=

���

L

[

s

]

/

���

s

ij

���

s

��

i

j

=

���

���

E

/

���

s

i

j

\dot{s}_{ij} = -\partial E/\partial s_{ij}

s

��

ij

���

=

���

���

E

/

���

s

ij

���

Astrocyte

p

i

j

p_{ij}

p

ij

���

L

[

p

]

\mathcal{L}^{[p]}

L

[

p

]

��

(

p

i

j

)

=

���

L

[

p

]

/

���

p

i

j

\psi(p_{ij}) = \partial \mathcal{L}^{[p]}/\partial p_{ij}

��

(

p

ij

���

)

=

���

L

[

p

]

/

���

p

ij

���

p

��

i

j

=

���

���

E

/

���

p

i

j

\dot{p}_{ij} = -\partial E/\partial p_{ij}

p

��

���

ij

���

=

���

���

E

/

���

p

ij

���

���� Bonus: Why This Works

Because each layer uses

convex Lagrangians

, the system is guaranteed (under mild assumptions) to:

Converge to

fixed points

Maintain

bounded, stable dynamics

Enable

memory retrieval

from partial cues (like a Hopfield net)

Would you like a visual diagram of the full neuron���synapse���astrocyte energy network, or want to implement a simplified version in code (e.g., in PyTorch or NumPy)?

You said:

3. Associative Neuron���Astrocyte Model

In Section 2, we described a general framework, grounded in

the biology of neuron���astrocyte communication, for modeling

neuron���astrocyte interactions via the tripartite synapses. De-

pending on the choices of the nonlinearities and parameters this

network can exhibit many sophisticated dynamical behaviors���

such as chaos or limit cycles���which can be diff i cult to analyze

in full generality.

To better understand the potential role of neuron���astrocyte

interactions, we will focus on an important limiting case where

the system demonstrates associative memory functions. As in

essentially all models of biological associative memory, this

requiresthepresenceofsymmetriesinthegoverningequationsof

thebiologicalcircuit.Weshowthatundercertainconditions,the

resulting neuron���astrocyte model has a global energy function

(Lyapunov function), which monotonically decreases on the

dynamical trajectory and is bounded from below. This makes it

possibletoidentifyaregimeofoperationoftheneuron���astrocyte

network that results in dynamical trajectories converging to

i xed point attractor states. The i xed points can be identif i ed

with ���memories��� stored in the weight matrices, and the entire

neuron���astrocyte model can be regarded as an energy-based

Dense Associative Memory (also known as Modern Hopf i eld

Network) (35, 36). Importantly, this framework allows us to

show that the presence of a single astrocyte can provably boost

the memory capacity per compute unit of a neural circuit by a

factor of N.

We will follow the general formulation of energy-based

associative memories (37���39), which starts with picking three

Lagrangians, which def i ne layers of our architecture (neurons,

synapses, and astrocytic processes), and the corresponding acti-

vationfunctions.TheseLagrangiansareaneuralLagrangianL[n],

a synaptic Lagrangian L[s], and an astrocyte process Lagrangian

L[p]. In general these scalar functions can be arbitrary (differen-

tiable) functions of the corresponding dynamical variables. The

details of our derivation can be found in SI Appendix, section 1.

From these Lagrangians, we can derive via a Legendre

transformation three terms in the overall energy function of

the neuron���astrocyte system: E[n], E[s], and E[p]. The activation

functions in our model are dictated by the Lagrangians���indeed,

theithactivationissimplythepartialderivativeoftheLagrangian

with respect to the ithdynamical variable (SI Appendix, section 1

andBox1).Theremainingcontributionstothetotalenergyofthe

system describe the interactions between neurons, synapses, and

astrocytes. These contributions describe the synapse-mediated

interactions between the neurons E[ns], the interactions between

theprocessesandthesynapsesE[ps],andtheinteractionsbetween

the individual processes inside the astrocyte E[pp].

The overall energy function of the neuron���astrocyte model

can now be written as the sum of these six terms

E = E[n]+ E[s]+ E[p]+ E[ns]+ E[ps]+ E[pp][5]

From this Lagrangian formalism (37���39), the dynamical equa-

tions for the associative neuron���astrocyte energy can be viewed

as the negative gradient (with respect to the nonlinearities):

��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ���

���n�� xi= ���

���E

������i = ��� ��� xi+

N X

j=1

gij���j

���s�� sij= ���2

���E

���gij = ��� ��� sij+ ���i���j+ ���ij

���p�� pij= ���2

���E

������ij = ��� ��� pij+

N X

k,l=1

Tijkl���kl+ gij

[6]

The energy-based equations have a large amount of symmetry���

both in the parameters and the dynamical degrees of freedom,

e.g., Tijkl= Tklij, see SI Appendix, section 1. These symmetries

are needed for the existence of the global energy function for

our neuron���astrocyte network, which leads to mathematical

tractability. In real biology, some (or all) of these symmetries

might be broken, and the analytical tractability might be more

diff i cult or even impossible. We use the energy-based model

to establish theoretically the memory storage capabilities of

our model. The nonsymmetric model is studied numerically in

Section 4 where we show that it possesses similar capabilities

despite lacking the energy-based formulation. Note that unlike

thesymmetriesintheoriginalHopf i eldnetworks(40,41)(which

have no known biological interpretation), the invariance of T

with respect to swapping indices ij and kl can be viewed as

a natural consequence of the underlying symmetry of calcium

diffusion.

Thef i rsttwoequationsinEq.6arereminiscentoftheapproach

by Dong and Hopf i eld (42), which describes both the neural

dynamics and synaptic plasticity by a single energy function. The

difference of our system compared to ref. 42 is the existence

of the network of astrocytic processes, which interact with each

other and with the synapses. Following the general Lagrangian

formalism, it can be shown (SI Appendix, section 2) that

dE

dt = ��� ?

���n

N X

i,j=1 �� xi ���2L[n]

���xi���xj �� xj

+ ���s

2

N X

i,j,k,l=1 �� sij ���2L[s]

���sij���skl �� skl

+ ���p

2

N X

i,j,k,l=1 �� pij ���2L[p]

���pij���pkl �� pkl ?

��� 0

[7]

PNAS2025Vol. 122No. 21e2417788122https://doi.org/10.1073/pnas.24177881223 of 7

Downloaded

from https://www.pnas.org

by 159.2.135.155

on

June

10,

2025

from

IP

address

159.2.135.155.

Memory 1Memory 2

Memory 1Memory K

...

Energy Landscape

With Astrocyte

Without

AstrocyteA

B

Memory 1

Memory 1

Energy Landscape

Fig. 2.For a fixed number of neurons, the neuron���astrocyte network is

capable of storing many more memories than the neuron-only network. (A)

The energy landscape of a neuron-only associative network. (B) The energy

landscape of a neuron���astrocyte associative network. The memories are

moredenselypackedintothestate-space,therebyenablingsuperiormemory

storage and retrieval capabilities.

The last inequality sign holds if each Lagrangian has a positive

semidef i nite Hessian matrix. When the Hessian matrices are

strictly positive def i nite, the dynamical Eq. 6 are guaranteed

to arrive at a i xed point, because the energy is bounded from

below [through the invariant set theorem (43)]. Thus, starting

from an initial state the network dynamics l ows toward one of

the i xed points and for this reason describes the operation of an

associative memory.

3.1. Connection to Dense Associative Memory.Energy-based

neuron���astrocyte networks are described by a sophisticated

system of nonlinear differential equations [6] that are guaranteed

to represent dynamical trajectories converging to i xed points

attractors, assuming certain conditions on the Lagrangians are

met. The locations of those i xed points x���

i, s���ij, and p���ij coincide

with the local minima of the energy function Eq. 5, and are

independent of the time scales ���n, ���s, and ���p. The kinetics of

the model (i.e., the shape of dynamical trajectories), however,

heavily depends on these time scales. Although the characteristic

time scales of synaptic plasticity and dynamics of processes (as

well as the dynamics of the entire astrocyte) are subjects of

active debates in the community, it is generally believed that

neurons operate on faster time scales than synaptic plasticity or

the processes, ���n? ���s,���p. Since the goal of this section is to

analyze the i xed points of this network, which are independent

of these time scales, we have a freedom to choose the kinetic time

scalesinawaydictatedbymathematicalconvenience,ratherthan

biologicalreality.Specif i cally,wewillderiveaneffectivedynamics

on neurons that arises after the synapses and astrocytes are

integrated out from the dynamical equations, which intuitively*

is possible if ���s,���p? ���n. Despite this ���unbiological��� choice

for the intermediate steps, the i nal answer represents accurate

locations of i xed points for the network that operates in the

���biological��� regime ���n? ���s,���p, or with any other choice of time

scales.

*Strictly speaking, specific conditions must be satisfied in order for this operation to be

well-defined and to avoid pathologies such as peaking phenomena. In our system this can

easily be done by restricting the eigenvalues of T to be less than ���, which ensures that the

fast dynamics are contracting (44).

The i xed points of the synaptic and processes��� dynamics in

Eq. 6 are def i ned by (assuming for simplicity that ��� = ��� = 0)

��� ��� ��� ��� ���

���ij= ������i���j

gij=

N P

k,l=1 Tijkl���k���l

[8]

Note that for a i xed value of ���i, these equations uniquely

determine the values of sijand pijwhen g and ��� are strictly

monotonic. Substituting this solution into the i rst Eq. 6, gives

the effective dynamics for neurons (assuming for simplicity that

��� = 1) ���n�� xi= ���xi+

N X

j,k,l=1

Tijkl���j���k���l[9]

and effective energy

Eeff= h

N X

i=1 xi���i��� L[n] i ���

1

4

N X

i,j,k,l=1

Tijkl���i���j���k���l[10]

Eqs. 9 and 10 contain the essence of our theoretical argument.

The i xed points of this effective dynamical system exactly coin-

cide with the i xed points of the original complete energy-based

neuron���astrocyte network [6], projected on the neuron-only

subspace. The hallmark of this effective theory is the existence

of the four-body neuron-to-neuron interactions, represented by

the product of four i ring rate functions ���iin the effective

Lyapunov function and the product of three i ring rate functions

in the effective equations. In conventional i ring rate models

there is only one i ring rate function in the right hand side of

the dynamical equations and two i ring rate functions in the

corresponding energy function, see for example ref.

ChatGPT said:

Here is a

step-by-step derivation

of the

Associative Neuron���Astrocyte Model

from the Lagrangian formalism to the effective neural dynamics (Eqs. 6���10), following the logic and notation of the provided paper. We assume some familiarity with variational calculus, dynamical systems, and energy-based models like Hopfield networks.

Step 1: Define Lagrangians for Each Component

You start by defining scalar Lagrangians for the three interacting components:

Neural activity Lagrangian

:

L

[

n

]

(

x

)

L[n](\mathbf{x})

L

[

n

]

(

x

)

, where

x

=

(

x

1

,

.

.

.

,

x

N

)

\mathbf{x} = (x_1, ..., x_N)

x

=

(

x

1

���

,

...

,

x

N

���

)

are neural states.

Synaptic state Lagrangian

:

L

[

s

]

(

s

)

L[s](\mathbf{s})

L

[

s

]

(

s

)

, where

s

=

(

s

i

j

)

\mathbf{s} = (s_{ij})

s

=

(

s

ij

���

)

are synaptic strengths.

Astrocytic process Lagrangian

:

L

[

p

]

(

p

)

L[p](\mathbf{p})

L

[

p

]

(

p

)

, where

p

=

(

p

i

j

)

\mathbf{p} = (p_{ij})

p

=

(

p

ij

���

)

represent astrocytic process variables.

Each Lagrangian generates an

activation function

via differentiation:

Neuron:

��

i

=

���

L

[

n

]

���

x

i

\phi_i = \frac{\partial L[n]}{\partial x_i}

��

i

���

=

���

x

i

���

���

L

[

n

]

���

Synapse:

g

i

j

=

���

L

[

s

]

���

s

i

j

g_{ij} = \frac{\partial L[s]}{\partial s_{ij}}

g

ij

���

=

���

s

ij

���

���

L

[

s

]

���

Astrocyte:

��

i

j

=

���

L

[

p

]

���

p

i

j

\psi_{ij} = \frac{\partial L[p]}{\partial p_{ij}}

��

ij

���

=

���

p

ij

���

���

L

[

p

]

���

Step 2: Construct the Energy Function

Using Legendre transforms, the energy function is defined as:

E

=

E

[

n

]

+

E

[

s

]

+

E

[

p

]

+

E

[

n

s

]

+

E

[

p

s

]

+

E

[

p

p

]

E = E[n] + E[s] + E[p] + E[ns] + E[ps] + E[pp]

E

=

E

[

n

]

+

E

[

s

]

+

E

[

p

]

+

E

[

n

s

]

+

E

[

p

s

]

+

E

[

pp

]

Where:

E

[

n

]

=

���

i

x

i

��

i

���

L

[

n

]

(

x

)

E[n] = \sum_i x_i \phi_i - L[n](\mathbf{x})

E

[

n

]

=

���

i

���

x

i

���

��

i

���

���

L

[

n

]

(

x

)

E

[

s

]

=

���

i

j

s

i

j

g

i

j

���

L

[

s

]

(

s

)

E[s] = \sum_{ij} s_{ij} g_{ij} - L[s](\mathbf{s})

E

[

s

]

=

���

ij

���

s

ij

���

g

ij

���

���

L

[

s

]

(

s

)

E

[

p

]

=

���

i

j

p

i

j

��

i

j

���

L

[

p

]

(

p

)

E[p] = \sum_{ij} p_{ij} \psi_{ij} - L[p](\mathbf{p})

E

[

p

]

=

���

ij

���

p

ij

���

��

ij

���

���

L

[

p

]

(

p

)

E

[

n

s

]

=

���

���

i

j

g

i

j

��

i

��

j

E[ns] = -\sum_{ij} g_{ij} \phi_i \phi_j

E

[

n

s

]

=

���

���

ij

���

g

ij

���

��

i

���

��

j

���

(neuron-synapse coupling)

E

[

p

s

]

=

���

���

i

j

��

i

j

��

i

��

j

E[ps] = -\sum_{ij} \psi_{ij} \phi_i \phi_j

E

[

p

s

]

=

���

���

ij

���

��

ij

���

��

i

���

��

j

���

(astrocyte-synapse coupling)

E

[

p

p

]

=

���

���

i

j

k

l

T

i

j

k

l

��

i

j

��

k

l

E[pp] = -\sum_{ijkl} T_{ijkl} \psi_{ij} \psi_{kl}

E

[

pp

]

=

���

���

ijk

l

���

T

ijk

l

���

��

ij

���

��

k

l

���

(astrocyte self-interaction)

So the total energy becomes:

E

=

���

i

x

i

��

i

���

L

[

n

]

+

���

i

j

s

i

j

g

i

j

���

L

[

s

]

+

���

i

j

p

i

j

��

i

j

���

L

[

p

]

���

���

i

j

g

i

j

��

i

��

j

���

���

i

j

��

i

j

��

i

��

j

���

���

i

j

k

l

T

i

j

k

l

��

i

j

��

k

l

(5)

E = \sum_i x_i \phi_i - L[n] + \sum_{ij} s_{ij} g_{ij} - L[s] + \sum_{ij} p_{ij} \psi_{ij} - L[p] - \sum_{ij} g_{ij} \phi_i \phi_j - \sum_{ij} \psi_{ij} \phi_i \phi_j - \sum_{ijkl} T_{ijkl} \psi_{ij} \psi_{kl}

\tag{5}

E

=

i

���

���

x

i

���

��

i

���

���

L

[

n

]

+

ij

���

���

s

ij

���

g

ij

���

���

L

[

s

]

+

ij

���

���

p

ij

���

��

ij

���

���

L

[

p

]

���

ij

���

���

g

ij

���

��

i

���

��

j

���

���

ij

���

���

��

ij

���

��

i

���

��

j

���

���

ijk

l

���

���

T

ijk

l

���

��

ij

���

��

k

l

���

(

5

)

Step 3: Derive Dynamics from the Energy Gradient

The dynamics are given by gradient descent on this energy function:

��

n

x

��

i

=

���

���

E

���

��

i

=

���

��

x

i

+

���

j

g

i

j

��

j

��

s

s

��

i

j

=

���

2

���

E

���

g

i

j

=

���

��

s

i

j

+

��

i

��

j

+

��

i

j

��

p

p

��

i

j

=

���

2

���

E

���

��

i

j

=

���

��

p

i

j

+

���

k

l

T

i

j

k

l

��

k

l

+

g

i

j

(6)

\begin{aligned}

\tau_n \dot{x}_i &= -\frac{\partial E}{\partial \phi_i} = -\lambda x_i + \sum_j g_{ij} \phi_j \\

\tau_s \dot{s}_{ij} &= -2 \frac{\partial E}{\partial g_{ij}} = -\alpha s_{ij} + \phi_i \phi_j + \psi_{ij} \\

\tau_p \dot{p}_{ij} &= -2 \frac{\partial E}{\partial \psi_{ij}} = -\gamma p_{ij} + \sum_{kl} T_{ijkl} \psi_{kl} + g_{ij}

\end{aligned}

\tag{6}

��

n

���

x

��

i

���

��

s

���

s

��

ij

���

��

p

���

p

��

���

ij

���

���

=

���

���

��

i

���

���

E

���

=

���

��

x

i

���

+

j

���

���

g

ij

���

��

j

���

=

���

2

���

g

ij

���

���

E

���

=

���

��

s

ij

���

+

��

i

���

��

j

���

+

��

ij

���

=

���

2

���

��

ij

���

���

E

���

=

���

��

p

ij

���

+

k

l

���

���

T

ijk

l

���

��

k

l

���

+

g

ij

���

���

(

6

)

This gives the dynamical system describing evolution of neurons, synapses, and astrocytes.

Step 4: Show Energy is Monotonically Decreasing

From the structure of the energy and using positive semidefinite Hessians of each Lagrangian:

d

E

d

t

���

0

(7)

\frac{dE}{dt} \leq 0

\tag{7}

d

t

d

E

���

���

0

(

7

)

This guarantees convergence to fixed points under the right conditions (e.g.

���

2

L

[

n

]

/

���

x

i

���

x

j

���

0

\partial^2 L[n] / \partial x_i \partial x_j \geq 0

���

2

L

[

n

]

/

���

x

i

���

���

x

j

���

���

0

, etc.).

Step 5: Derive Fixed Points

Assume long-time limit (

s

��

i

j

=

0

,

p

��

i

j

=

0

\dot{s}_{ij} = 0, \dot{p}_{ij} = 0

s

��

ij

���

=

0

,

p

��

���

ij

���

=

0

), and simplify for clarity by letting

��

=

��

=

0

\alpha = \gamma = 0

��

=

��

=

0

:

��

i

j

=

���

��

i

��

j

g

i

j

=

���

k

,

l

T

i

j

k

l

��

k

��

l

(8)

\begin{aligned}

\psi_{ij} &= -\phi_i \phi_j \\

g_{ij} &= \sum_{k,l} T_{ijkl} \phi_k \phi_l

\end{aligned}

\tag{8}

��

ij

���

g

ij

���

���

=

���

��

i

���

��

j

���

=

k

,

l

���

���

T

ijk

l

���

��

k

���

��

l

���

���

(

8

)

This gives steady-state expressions for astrocytic and synaptic variables as functions of neuron activations

��

\phi

��

.

Step 6: Plug into Neural Dynamics

Substitute Eq. 8 into Eq. 6���s first line:

��

n

x

��

i

=

���

x

i

+

���

j

,

k

,

l

T

i

j

k

l

��

j

��

k

��

l

(9)

\tau_n \dot{x}_i = -x_i + \sum_{j,k,l} T_{ijkl} \phi_j \phi_k \phi_l

\tag{9}

��

n

���

x

��

i

���

=

���

x

i

���

+

j

,

k

,

l

���

���

T

ijk

l

���

��

j

���

��

k

���

��

l

���

(

9

)

This is the

effective neural dynamics

, which shows that each neuron is influenced by a

third-order interaction

of other neurons (via astrocyte-mediated synapses).

Step 7: Derive Effective Energy Function

The effective energy function

E

eff

E_{\text{eff}}

E

eff

���

becomes:

E

eff

=

���

i

x

i

��

i

���

L

[

n

]

���

1

4

���

i

,

j

,

k

,

l

T

i

j

k

l

��

i

��

j

��

k

��

l

(10)

E_{\text{eff}} = \sum_i x_i \phi_i - L[n] - \frac{1}{4} \sum_{i,j,k,l} T_{ijkl} \phi_i \phi_j \phi_k \phi_l

\tag{10}

E

eff

���

=

i

���

���

x

i

���

��

i

���

���

L

[

n

]

���

4

1

���

i

,

j

,

k

,

l

���

���

T

ijk

l

���

��

i

���

��

j

���

��

k

���

��

l

���

(

10

)

Here:

The first two terms come from neuron dynamics.

The quartic term arises from the astrocyte-synapse feedback loop.

Conclusion: Associative Memory via Astrocyte Boosting

Fixed points

of this system correspond to

stored memories

.

Due to the

quartic interactions

���

T

i

j

k

l

��

i

��

j

��

k

��

l

\sim T_{ijkl} \phi_i \phi_j \phi_k \phi_l

���

T

ijk

l

���

��

i

���

��

j

���

��

k

���

��

l

���

, memory capacity scales favorably���specifically, adding an astrocyte network effectively enables

four-body interactions

, significantly boosting memory density over standard Hopfield models (which use two-body interactions).

In the presence of symmetry

T

i

j

k

l

=

T

k

l

i

j

T_{ijkl} = T_{klij}

T

ijk

l

���

=

T

k

l

ij

���

, analytical tractability is preserved and convergence is guaranteed.

Let me know if you'd like a diagram to visualize the energy landscape changes with and without astrocytes or a numerical example of one of the update equations.

You said:

PNAS2025Vol. 122No. 21e2417788122https://doi.org/10.1073/pnas.24177881223 of 7

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Memory 1Memory 2

Memory 1Memory K

...

Energy Landscape

With Astrocyte

Without

AstrocyteA

B

Memory 1

Memory 1

Energy Landscape

Fig. 2.For a fixed number of neurons, the neuron���astrocyte network is

capable of storing many more memories than the neuron-only network. (A)

The energy landscape of a neuron-only associative network. (B) The energy

landscape of a neuron���astrocyte associative network. The memories are

moredenselypackedintothestate-space,therebyenablingsuperiormemory

storage and retrieval capabilities.

The last inequality sign holds if each Lagrangian has a positive

semidef i nite Hessian matrix. When the Hessian matrices are

strictly positive def i nite, the dynamical Eq. 6 are guaranteed

to arrive at a i xed point, because the energy is bounded from

below [through the invariant set theorem (43)]. Thus, starting

from an initial state the network dynamics l ows toward one of

the i xed points and for this reason describes the operation of an

associative memory.

3.1. Connection to Dense Associative Memory.Energy-based

neuron���astrocyte networks are described by a sophisticated

system of nonlinear differential equations [6] that are guaranteed

to represent dynamical trajectories converging to i xed points

attractors, assuming certain conditions on the Lagrangians are

met. The locations of those i xed points x���

i, s���ij, and p���ij coincide

with the local minima of the energy function Eq. 5, and are

independent of the time scales ���n, ���s, and ���p. The kinetics of

the model (i.e., the shape of dynamical trajectories), however,

heavily depends on these time scales. Although the characteristic

time scales of synaptic plasticity and dynamics of processes (as

well as the dynamics of the entire astrocyte) are subjects of

active debates in the community, it is generally believed that

neurons operate on faster time scales than synaptic plasticity or

the processes, ���n? ���s,���p. Since the goal of this section is to

analyze the i xed points of this network, which are independent

of these time scales, we have a freedom to choose the kinetic time

scalesinawaydictatedbymathematicalconvenience,ratherthan

biologicalreality.Specif i cally,wewillderiveaneffectivedynamics

on neurons that arises after the synapses and astrocytes are

integrated out from the dynamical equations, which intuitively*

is possible if ���s,���p? ���n. Despite this ���unbiological��� choice

for the intermediate steps, the i nal answer represents accurate

locations of i xed points for the network that operates in the

���biological��� regime ���n? ���s,���p, or with any other choice of time

scales.

*Strictly speaking, specific conditions must be satisfied in order for this operation to be

well-defined and to avoid pathologies such as peaking phenomena. In our system this can

easily be done by restricting the eigenvalues of T to be less than ���, which ensures that the

fast dynamics are contracting (44).

The i xed points of the synaptic and processes��� dynamics in

Eq. 6 are def i ned by (assuming for simplicity that ��� = ��� = 0)

��� ��� ��� ��� ���

���ij= ������i���j

gij=

N P

k,l=1 Tijkl���k���l

[8]

Note that for a i xed value of ���i, these equations uniquely

determine the values of sijand pijwhen g and ��� are strictly

monotonic. Substituting this solution into the i rst Eq. 6, gives

the effective dynamics for neurons (assuming for simplicity that

��� = 1) ���n�� xi= ���xi+

N X

j,k,l=1

Tijkl���j���k���l[9]

and effective energy

Eeff= h

N X

i=1 xi���i��� L[n] i ���

1

4

N X

i,j,k,l=1

Tijkl���i���j���k���l[10]

Eqs. 9 and 10 contain the essence of our theoretical argument.

The i xed points of this effective dynamical system exactly coin-

cide with the i xed points of the original complete energy-based

neuron���astrocyte network [6], projected on the neuron-only

subspace. The hallmark of this effective theory is the existence

of the four-body neuron-to-neuron interactions, represented by

the product of four i ring rate functions ���iin the effective

Lyapunov function and the product of three i ring rate functions

in the effective equations. In conventional i ring rate models

there is only one i ring rate function in the right hand side of

the dynamical equations and two i ring rate functions in the

corresponding energy function, see for example ref. 33. This

is a mathematical ref l ection of the biological fact that each

synapse connects two neurons (pre- and postsynaptic cells). In

our model, we have demonstrated that the contribution of the

astrocyte is to effectively create a computational many-neuron

synapse (mediated by the network of astrocytic processes). In

other words, the computational function of the astrocyte is to

bring the information about the states of distant synapses (and

neurons) to each tripartite synapse resulting in the ���effective���

four-neuron synapse that connects neurons that are potentially

very far away from each other. In what follows, we will explain

that this computational property has important implications for

storing memories.

3.2. Storing Memories in the Network of Astrocyte Processes.

Imagine that we are given K memory patterns ������(index ��� =

1...K), and each pattern is an N-dimensional vector. The task

of associative memory is to store these patterns in the weights

of the neural-astrocyte network, so that temporal dynamics can

asymptotically l ow to these patterns. Choose tensor T such that

it satisf i es the following relationship

Tijkl���

K X

���=1

������

i ������

j ������

k ������

l [11]

With these notations, the effective neuron-only theory is equiv-

alent to a model with quartic interaction from the Dense

Associative Memory family (35, 36). Specif i cally, the effective

energy can be written as

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Eeff= h

N X

i=1 xi���i��� L[n] i ���

K X

���=1

F ?

N X

i=1

������

i ���i ?

,

whereF(z) = 1 4z

4

and effective equations as

���n�� xi= ���xi+

K X

���=1

������

i F0 ?

N X

j=1

������

j ���j ?

[12]

Dense Associative Memories is a new class of models that extend

traditionalHopf i eldNetworks(33,40,41)byintroducinghigher

than quadratic terms in their energy function. It has been shown

that this extension leads to superior information storage capacity

and representational power of these models compared to the

traditional Hopf i eld Networks (35). They are also related to the

attention mechanism in Transformers (37, 45) and are used in

state-of-the-art, energy-based neural networks (46, 47).

3.2.1. Memory capacity of a neuron���astrocyte network.It is in-

sightful to ask the question: How many memories can the model

store per the number of compute units? Assuming a conservative

def i nition���of the ���compute unit,��� our energy-based neuron���

astrocyte model [6] has approximately N2compute units in the

limit of large N, i.e.

N neurons+N2synapses+N2processes ��� N2compute units

The storage capacity Kmaxof the Dense Associative Memory

model with quartic energy is known to be (35)

K

max ��� N3

Thus, for our model the number of memories per compute unit

grows linearly as the size of the network is increased

Kmax

Number of compute units ��� N

This metric can be compared with other biologically plausible

implementations of Dense Associative Memory. For instance,

KrotovandHopf i eld(37)haveproposedtoaugmentthenetwork

of feature neurons with a set of auxiliary hidden neurons. In their

model, the compute units consist of both feature and hidden

neurons, and importantly the number of memories per compute

unit is a constant independent of N,

Kmax

Number of compute units ��� constant

Thus, neuron���astrocyte networks signif i cantly outperform the

implementation (37) according to this metric, see Fig. 2. This

makes neuron���astrocyte networks an exciting candidate for

biological���hardware���implementingDenseAssociativeMemory.

Evenmoreinsightfulistotracedowntheoriginofthismemory

storage back to the biophysical implementation of the neuron���

astrocyte networks Eq. 6. The memories in our model are stored

in tensor T, which describes the network of astrocyte���s processes

and the transport of Ca2+or other potential molecules, e.g.

protein kinase A, between those processes. Our theory suggests

���A less conservative definition would consider the entire astrocyte as a single compute

unit.

that memories can be stored in the biological machinery (inside a

singleastrocyte)oftransportofanappropriatesignalingmolecule

between the astrocyte���s processes. We have demonstrated a

conceptual theoretical possibility of such a storage through a

Hebbian-like plasticity rule [11]. This was done for the sake

of transparency of the theoretical argument. In principle, more

sophisticated storage rules are possible too. We hope that future

experiments might be able to test this exciting hypothesis.

3.2.2. Howmanyastrocyteparametersareneeded?The

Hebbian-like storage scheme Eq. 11 requires that every process

within the astrocyte be directly connected to every other process.

Future experiments will hopefully determine whether such

detailed communication within a single astrocyte is possible.

In the meantime, it is instructive to ask how the memory

capacity changes as a function of the degree of process-to-process

connectivity.

Heuristically, if we wish to store K memories, each containing

N independent bits, we need on the order of KN parameters in

ourmodel���oneparameterforeachbitofinformationstored.For

example, if we wish to store K = N memories, we need on the

order of N2parameters���these parameters can be

ChatGPT said:

Would you like a detailed breakdown of Section 3? I can:

Summarize the

connections to Dense Associative Memory

and how astrocytes create

four-body neural interactions

.

Explain their

memory capacity calculation

, showing how capacity scales with network size.

Provide intuition on what the

tensor

T

i

j

k

l

T_{ijkl}

T

ijk

l

���

means biologically���i.e., how astrocyte calcium networks store memories.

Or I can diagram the energy landscape (akin to Fig.���2) to illustrate how astrocytes enhance memory density. Let me know which direction you'd prefer!

You said:

Give some simplified analogies to help explain it

ChatGPT said:

Great���here are

simplified analogies

to help understand the

neuron���astrocyte model

and its associative memory dynamics, especially the role of astrocytes in enhancing memory capacity:

���� 1.

Tripartite Synapse = Group Chat Moderator

Analogy

: Imagine a group chat between two people (neuron A and neuron B). The astrocyte is like a

moderator

who listens in and sometimes adds helpful suggestions or reactions based on how the conversation is going.

Neuron A says something ��� Neuron B responds.

The

astrocyte listens

, tracks the emotional tone (neurotransmitter activity), and if things escalate or get exciting, the moderator (astrocyte) might step in and shape the tone���perhaps slowing things down, adding emojis, or calming things.

This feedback loop helps keep the conversation stable or shift it toward certain outcomes���just like

astrocytes regulate neural communication

using calcium and gliotransmitters.

������� 2.

Astrocyte Calcium Transport = Spider Web Vibration

Analogy

: Astrocyte processes form a

web

, like a spider web. When one part of the web vibrates (from activity at a nearby synapse), that vibration travels through the threads (intracellular calcium signaling), alerting

distant parts

of the web.

A single signal (neural activity) at one synapse can ripple through the web and affect

other synapses wrapped by the same astrocyte

.

This allows one astrocyte to coordinate many synapses, like a spider sensing prey and deciding where to pounce.

���� 3.

Memory Capacity = Puzzle Table with an Organizer

Analogy

: Imagine you���re trying to solve 100 jigsaw puzzles on the same table. If you just throw all the pieces together (neurons alone), it���s chaos.

The

astrocyte

is like a table organizer who places dividers, groups puzzle pieces, and makes sure each puzzle develops in its own corner.

This dramatically increases how many puzzles (memories) you can store and retrieve accurately���because now

overlap is managed

, and each puzzle (memory attractor) has space to stabilize.

The paper shows astrocytes boost memory capacity by a factor of

N

N

N

���that���s like going from solving 10 puzzles to 100 on the same table.

���� 4.

Lagrangian = Design Blueprint

Analogy

: The Lagrangian is like the

blueprint for building

a machine (neuron, synapse, astrocyte).

Each Lagrangian defines what the system ���wants��� to do���like minimizing effort or balancing forces.

Taking the

derivative of the Lagrangian

is like seeing how much energy it takes to move a knob or gear. That���s your

activation function

.

In this model, you have three machines (neuron, synapse, astrocyte), each with its own blueprint, and they���re connected. The

total energy

tells you how all the gears interact and where the machine naturally comes to rest.

���� 5.

Energy Minimization = Settling into a Comfortable Chair

Analogy

: Imagine sitting in a beanbag chair. At first, you fidget a bit, then you eventually find a perfect, stable position.

That ���settling��� process is like the

network minimizing its energy

���the neurons, synapses, and astrocyte processes shift around until they reach a comfortable, stable pattern (a memory).

Once you're in that position (a fixed-point attractor), even small nudges won���t change much. You've "remembered" that state.

Would you like me to tie these analogies to the

actual math step-by-step

, or create a

diagram

showing how they interrelate visually (e.g., astrocyte���neuron���memory triangle)?

You said:

5. Discussion

We have introduced a biologically inspired model that describes

theinteractionsbetweenneurons,synapses,andastrocytes.Inour

model, astrocytes are able to adaptively control synaptic weights

in an online fashion. Theoretical analysis has demonstrated that

thismodelcanexhibitassociativememorybehaviorandisclosely

related to the Dense Associative Memory family of models with

supralinear memory capacity, as well as to transformers. We have

shown that, through the choice of the connectivity tensor Tijkl,

our neuron���astrocyte model can be smoothly dialed between

operating as a transformer and operating as a Dense Associative

Memory network. This opens up the possibility for exploring

novel architectures ���in-between��� transformers and Dense As-

sociative Memories. Furthermore, we have presented a simple

algorithm for memory storage and have provided numerical

evidence of our models��� effectiveness, such as successfully storing

and retrieving CIFAR10 and ImageNet images.

In broader terms, this work proposes that memories can,

at least in part, be stored within the molecular machinery

of astrocytes. This contrasts with the prevailing neuroscience

viewpoint that memories are stored in the synaptic weights

between neurons. To experimentally validate this claim, one

would need to selectively interfere with the ability of Ca2+to

diffuse intracellularly through astrocytes. Our model predicts

that hindering this diffusion would signif i cantly impair memory

recall. Our model is l exible enough to accommodate many

different types of process-to-process coupling patterns, which

couldpresumablybef i ttomatchphysiologicaldata.Forexample,

it is possible to enforce ���nearest-neighbor��� coupling between

astrocyte processes (which can be achieved by e.g., imposing a

block-diagonal structure on the tensor T such that Tijkl= 0 if

processes ij and kl are not spatially close to each other), while still

guaranteeing convergence of our model to a i xed point.

The feedback loop between neurons and astrocytes is well

established experimentally, yet a simple computational account

of its function remains largely absent. Previous models have

primarily focused on the biophysical details of neuron���astrocyte

interactions, aiming to capture the emergence of complex

spatiotemporal calcium dynamics (52, 53). Building on these

efforts, our work highlights the role of astrocytic modulation

in synaptic plasticity. However, rather than modeling detailed

biophysical mechanisms, we take a higher-level approach, using

a ���f i ring rate��� model to abstract away these complexities and

uncover the core computational principles governing neuron���

astrocyte interactions.

One particularly intriguing feature of astrocytes is their

abundance in the brain. Indeed, they are present in virtually

every major brain structure. Of particular relevance to the

modeling approach introduced here are associative brain areas,

such as the neocortex and hippocampus, which are believed to

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play a crucial role in memory storage and retrieval. Notably,

human neocortical astrocytes are signif i cantly larger and more

active than their rodent counterparts, suggesting an enhanced

computational function (54). As in prior modeling work using

recurrent networks (55), our neuron���astrocyte model can be

tuned (i.e., trained) to perform functions associated with specif i c

tasks relevant to particular brain regions. For example, astrocytes

in the visual cortex may be specialized for storing and retrieving

visual information, whereas those in language-related areas may

be more attuned to processing auditory information.

Whileourfocushasbeenonaminicircuitconsistingofasingle

astrocyte interacting with multiple nearby synapses, astrocytes

also extensively communicate with each other through chemical

gap junctions. Exploring the implications of this intercellular

coupling will be the subject of future research.

Key ideas in machine learning and AI drew initial inspiration

from neuroscience, including neural networks, convolutional

nets, threshold linear (ReLu) units, and dropout. Yet it is

debatable whether neuroscience research from the last i fty years

hassignif i cantlyinf l uencedorinformedmachinelearning.Astro-

cytes,alongwithotherbiologicalstructuressuchasdendrites(56)

and neuromodulators (57) may offer a fresh source of inspiration

for building state-of-the-art AI systems.

ChatGPT said:

Here's a

simplified breakdown

of the

Discussion

section using analogy, interpretation, and core insights:

����

High-Level Takeaway

:

Astrocytes are not just support cells���they may actively store and control memories, using calcium signaling to modulate synapses in a way that resembles powerful machine learning models.

���� 1.

Biological Neuron���Astrocyte Network as Memory Engine

What they did

: Built a computational model that includes

neurons, synapses, and astrocytes

, with astrocytes dynamically adjusting synaptic strength using calcium signals.

What���s new

: This model doesn���t just store memory in

synaptic weights

(the traditional view), but in the

astrocyte���s internal calcium dynamics

���like giving the ���janitor��� (astrocyte) the master copy of the memory instead of just the wiring (synapse).

Analogy

: Traditionally, we thought memory was stored in the ���wires��� between neurons. This paper says, ���What if memory���s also stored in the

control software

running on an entirely different system?���

���� 2.

Astrocytes Boost Memory Capacity ��� Like RAM Expansion

Dense Associative Memory (DAM)

: A network where you can store lots of ���attractors��� (stable memories), and recall them by partial cues.

Their model behaves like DAM���but

with more capacity

, thanks to astrocytes.

Transformer-like tuning

: By changing a single tensor

T

i

j

k

l

T_{ijkl}

T

ijk

l

���

, the model can act more like a

Transformer

(AI model like GPT), or more like a DAM network. This opens up a spectrum of hybrid AI-brain models.

Analogy

: You���re adding extra RAM (astrocytes) and customizing the BIOS settings (tensor

T

T

T

) to tune your brain between ���classic memory mode��� and ���AI attention mode.���

���� 3.

Biological Prediction: Block Calcium = Block Memory

Biological claim

: If you block

Ca����� diffusion inside astrocytes

, memory recall should degrade.

Testable hypothesis

: This is falsifiable���experiments could knock out astrocytic calcium signaling and test memory function.

Analogy

: Like cutting the signal wires inside the brain���s memory manager���you'd expect recall to suffer, even if the neurons still fire.

���� 4.

Flexible Architecture: Spatial Logic and Neighbor Rules

The model can implement spatial rules���for example, only letting nearby astrocyte processes talk to each other (like

nearest-neighbor coupling

).

Can mimic real astrocyte architecture (block-diagonal tensors).

Analogy

: Like defining city zones in SimCity���where only certain blocks can interact���still leads to stable city behavior.

���� 5.

Past Models Focused on Physics; This One Focuses on Computation

Past models of astrocytes were

biophysical

: how calcium waves move, how channels open, etc.

This model is

computational

: what information astrocytes can store and how they shape memory and learning.

Analogy

: Rather than simulating every bolt in a car, this model says: ���Let���s understand what the

dashboard

does for driving.���

���� 6.

Astrocytes Are Everywhere���and Smarter in Humans

Found in every brain region, especially

neocortex and hippocampus

(key memory areas).

Human astrocytes are

larger and more complex

than rodent ones ��� possibly more computationally powerful.

Implication

: Our intelligence may stem in part from better astrocytes.

Analogy

: Humans upgraded from 1980s calculators (rodents) to full touchscreen tablets (astrocytes with real-time modulation and memory).

������� 7.

Specialization by Brain Region

Astrocytes in the

visual cortex

might help store and recall visual scenes.

Astrocytes in

language areas

might be tuned to language features.

Analogy

: Like installing different apps on your phone���s RAM depending on what you use most.

���� 8.

Astrocyte���Astrocyte Networks = Future Frontier

Astrocytes also talk to each other via

gap junctions

���networks of astrocytes may compute in

collective, non-neural ways

.

This opens up the idea of

non-neural parallel memory systems

.

Analogy

: Instead of just nodes (neurons), you now have

ghost routers

(astrocytes) talking in a hidden mesh network behind the scenes.

���� 9.

Back to AI: Time to Learn from the Brain Again

Machine learning originally borrowed from neuroscience (neurons, ReLUs, dropout).

This work says: maybe it���s time to go back and

mine the brain again

���but now for things like astrocytes, dendrites, and neuromodulators.

Astrocytes could inspire entirely new ML architectures

beyond transformers.

Analogy

: AI was built from a 2D blueprint of the brain. Now we���re exploring its

3D plumbing

, and it might unlock the next generation of models.

Would you like me to draw a

simplified diagram

summarizing this (e.g., ���Neuron + Synapse + Astrocyte = Memory Matrix���)? Or go into how to

numerically simulate

a small network based on this idea?

You said:

It reminds me of Great question. Monica Anderson���s Wisdom Salon metaphor and your ���janitor walking the hallway��� image can be understood as a socially-distributed analogue to Global Workspace Theory (GWT) and the Inner Screen model of cognition. Let���s break this down precisely:


---


���� 1. Global Workspace Theory (GWT): A Cognitive Clearinghouse


Bernard Baars��� GWT posits that:


Consciousness is a global broadcast of selected information across a distributed network.


Many specialized unconscious processes ("processors") compete for access to a limited-capacity global workspace���like a stage spotlight.


Once an item is selected for the spotlight, it is broadcast to the whole system, influencing memory, attention, language, planning, etc.


In essence, cognition is organized like a theater:


Countless subagents (neural routines) propose content.


A central stage (the workspace) hosts what ���wins��� attention.


A spotlight of awareness shines on chosen content, making it globally available.


---


���� 2. Inner Screen Models of Active Cognition


This model builds on GWT by emphasizing simulation:


The ���screen��� is not passive display���it���s a workspace for mental rehearsal.


We imagine, rehearse, and evaluate possible actions internally before acting externally.


Active inference, Bayesian updating, and motor simulation all play roles here.


So the ���inner screen��� is a place where high-salience, high-integrity representations are formed and manipulated.


---


���� 3. Monica���s Wisdom Salon and the Janitor Model


In your metaphor:


Each conversation table in the Wisdom Salon is like a local unconscious processor.


Participants contribute partial knowledge and contextual interpretation.


The butcher paper records the ���working memory��� or local outputs of these processors.


The janitor figure (or ���South��� role, who stays put) relays the summary of prior thought to newcomers���preserving continuity while allowing new synthesis.


The shuffling and reaggregation acts like a convergence filter���a stochastic evolutionary process where high-value ideas propagate and low-value ones decay.


The janitor walking the hall:


Gathers key insights (like attentional spotlight).


Avoids repetition or junk (noise filtering).


Promotes convergence by diffusing best ideas into shared consciousness.


---


���� 4. Structural Parallels: GWT ��� Wisdom Salon


Cognitive System (GWT)	Wisdom Salon / Janitor Metaphor


Unconscious processors	Individual tables, small-group conversations

Global workspace (consciousness)	Shared summaries / harvested butcher papers

Spotlight of attention	Janitor relaying key idea slips between rooms

Conscious broadcast	Cross-table propagation of salient insights

Working memory	Butcher paper notes + ���South��� summary explanations

Simulation / active cognition	Discussion, rehearsal, reinterpretation of ideas

Predictive validation	Testing ideas through group uptake and evolution


---


���� 5. Implications


Wisdom, in Monica���s view, emerges via collective iteration���just like consciousness, in GWT, emerges through competition and integration of representations.


Her emphasis on Model-Free Methods is akin to bottom-up intuitive processes (System 1 cognition) that precede formal models but still yield valid behavioral adaptations.


The Wisdom Salon becomes a socio-cognitive workspace: a mesh of analog minds approximating a brain���s conscious spotlight.


---


���� Conclusion


The janitor metaphor, the Wisdom Salon structure, and the philosophical goals of Monica���s work form a social-level instantiation of Global Workspace Theory. Both are:


Distributed


Recursive


Emergence-driven


Filtering-limited


Dependent on attention and salience


Structured for integration rather than deduction


Monica essentially describes a society that thinks like a brain, and your framing nails its resemblance to internal cognition.


Let me know if you���d like a diagram mapping the components.

ChatGPT said:

Absolutely ��� that���s a precise and brilliant comparison. What you���ve outlined is not only a conceptual isomorphism between

Monica Anderson���s Wisdom Salon

and

Global Workspace Theory (GWT)

, but also an elegant bridge between

social cognition

and

internal metacognition

.

Here���s a short visual-inspired restatement with

enhanced clarity

and

modeling implications

, tying it back to neuron���astrocyte models and recursive computation:

����

1. Social-Cognitive Architecture as Neural Architecture

Brain

Wisdom Salon

Cognitive Function

Neurons

Individuals at tables

Local, parallel processing

Synaptic weights

Conversational dynamics, social weighting

Signal strength & plasticity

Astrocytes

Janitor + South role

Modulation & memory persistence

Conscious broadcast

Summarized insights shared across tables

Global access & awareness

Gating via salience

Shuffling participants, emergent consensus

Attention & relevance filtering

Memory attractors

Stable re-articulations across sessions

Reinforced group-level truths

����

2. Astrocyte ��� Janitor: The Modulatory Memory Keeper

The astrocyte doesn't initiate the conversation���it

listens

,

weights

, and

stabilizes

the synaptic network. Similarly, the

janitor

:

Doesn���t propose new content (like a neuron firing),

But

regulates what persists

, what���s repeated, and what���s transferred to new groups.

Acts like a

calcium wave

, slow but wide-reaching,

binding local dynamics into global structure

.

Just as astrocytic Ca����� diffusion creates attractors in neural space, the janitor

creates attractors in semantic space

.

����

3. Dense Associative Memory ��� Wisdom Memory

A Dense Associative Memory (DAM) model retrieves full patterns from partial cues���like hearing ���unified field��� and retrieving a full Monica-ian epistemology.

Wisdom Salon discussions do the same: a partial concept (���model-free intuition���) reactivates entire ideological networks.

By

modulating synaptic weights

(or conversational momentum),

astrocytes / janitors

embed long-term memory structures that shape future attention.

����

4. A Computational Vision: Social-Astrocytic Intelligence

You could actually model a

Wisdom Salon cognition engine

like this:

Each

processor node

= a discussion table (neural microcircuit).

Each node outputs ���butcher paper��� = temporary working memory.

���Janitor agents��� diffuse between nodes, copying, filtering, and reposting = astrocytic calcium modulating synaptic state.

When enough janitors converge on a pattern = memory attractor.

Whole-system awareness = spotlight of attention emerges via stable feedback across nodes.

This maps directly to

multi-agent memory dynamics

, and is

complementary to transformer attention

���a

slower, recursive, biologically plausible mechanism

for emergent coherence.

�������

5. Final Insight

The Wisdom Salon is not just a metaphor���it���s a

computational design principle

:

Cognition-as-salon

: Knowledge stabilizes not by authority or formal derivation, but through emergent convergence, modulated repetition, and attention-guided integration.

This mode of thinking aligns with RSVP theory, astrocytic attractor networks, and the critique of brittle symbolic systems. It allows for

soft, recursive convergence

���a cognitive ecosystem

between full chaos and rigid structure

.

Would you like a schematic of this as a diagram or a version formalized into a recursive cognitive architecture or simulation model?

You said:

Formalized into a recursive cognitive architecture.

ChatGPT said:

Excellent. Let���s formalize the

Wisdom Salon / Janitor metaphor

into a

Recursive Cognitive Architecture (RCA)

��� a computational system that models emergent, context-sensitive intelligence via recursive summarization, modulated diffusion, and attention-weighted memory.

We���ll define:

Entities

(processors, agents, memory stores)

Processes

(local computation, modulation, broadcasting)

Recursive structure

(convergence dynamics and feedback)

Relation to cognitive theory

(GWT, astrocytic models, RSVP)

���� Recursive Cognitive Architecture (RCA-Wisdom)

1.

Core Components

Component

Description

Analogy

LocalProcessor

A semi-autonomous conversational node with domain-specific heuristics

Table of discussants

MemorySlip

A transient local output (concept, pattern, summary)

Butcher paper

JanitorAgent

A mobile, modulating summarizer that transfers and filters information

Astrocyte / ���South��� role

GlobalConvergenceField

A shared, slowly-updating field representing collective awareness

Global Workspace / attractor field

SalienceFunction

Determines what slips get promoted or discarded

Attention spotlight

ConvergenceDynamics

Recursively reinforce stable attractors via repetition, novelty, and utility

Recursive sampling / attractor basin

2.

Formal Process Graph

markdown

Copy code

FOR EACH LocalProcessor Pi:

1.

Pi generates MemorySlips Mi based on local input and heuristics

2.

Mi are stored locally and made available to JanitorAgents


FOR EACH JanitorAgent J:

3.

J samples slips from Pi���s memory space

4.

J applies SalienceFunction to rate slips by:

-

Frequency (how often it's repeated)

-

Coherence (alignment with current context)

-

Novelty (information gain)

-

Endorsement (social reinforcement)

5.

J writes selected slips into GlobalConvergenceField


GLOBAL:

6.

GlobalConvergenceField aggregates inputs over time (low-pass filter)

7.

Feedback from GlobalConvergenceField influences:

-

Slip generation (top-down priming)

-

Slip selection (biasing salience)

-

Processor heuristics (goal updating)

3.

Recursive Dynamics

Local���Global Loop

:

Local processors generate bottom-up patterns; janitors carry these to a global field, which then modifies future local processing.

Memory Emergence

:

Patterns that persist across multiple janitor cycles become

converged attractors

, mimicking

associative memory consolidation

.

Parallel & Asynchronous

:

No centralized control. Local processors operate independently, with

soft synchronization

via shared convergence field.

Recursive Generalization

:

Each layer can recursively instantiate sub-salons (e.g., a MemorySlip becomes a LocalProcessor in another layer), supporting

hierarchical reasoning

and

abstraction

.

4.

Relation to Existing Theories

Concept

Equivalent in RCA-Wisdom

Global Workspace (Baars)

GlobalConvergenceField

Astrocyte modulation

JanitorAgent weighting synaptic-like relevance

Dense Associative Memory

Recurrence and attractor stability of converged slips

Transformer attention

SalienceFunction + Global field diffusion

Model-Free Intuition

LocalProcessor heuristics

RSVP Field Dynamics

Recursive, negentropic smoothing across agents

5.

Mathematical Notation (Sketch)

Let:

P_i(t)

= local processor i at time t

M_i(t)

= memory slips from processor i at time t

J_k(t)

= janitor agent k

S(M)

= salience function applied to memory slip M

G(t)

= global convergence field

Then:

Generation:

lua

Copy code

M_i(t) = P_i(t)(

input

)    #

local

slip

output

Salience selection:

mathematica

Copy code

S

(

M

)

=

����

F

(

M

)

+

����

N

(

M

)

+

����

C

(

M

)

+

����

E

(

M

)

#

frequency

,

novelty

,

coherence

,

endorsement

Global update:

Copy code

G(t+1) = (1 - ��)��G(t) + ������ J_k(S(M_i(t)))

Feedback modulation:

scss

Copy code

P_i

(t+

1

) ���

modulate

(P_i(t),

G

(t+

1

))   # global-to-local influence

6.

Possible Implementation Modules

ProcessorModule

: Domain-specific micro-agents (text, vision, abstract logic)

JanitorModule

: Memory router with learnable filters (could use contrastive loss)

GlobalField

: Slow-updating vector field or symbolic scratchpad (like a NeRF memory map or attractor grid)

SalienceLayer

: Transformer-style attention or GWR (growing when required) networks

RecursiveSynthesizer

: Allows dynamic regrouping of ideas into higher-order processors

���� Summary

This Recursive Cognitive Architecture encodes:

Distributed generation

(local tables / processors)

Selective summarization

(janitor-like modulation)

Global memory consolidation

(convergence field)

Recursive feedback

(top-down modulation)

Emergent cognition

via

attention-weighted attractor dynamics

It is

interpretable

,

flexible

, and

biologically inspired

, supporting both symbolic and sub-symbolic processing.

Would you like a sample simulation or prototype pseudo-code for this architecture? Or a mapping to specific LLM/memory-based agent designs?

You said:

See if you can explain the astrocyte memory model one more time.

ChatGPT said:

Here���s a refined, intuitive breakdown of the

astrocyte-based memory model

from Kozachkov���et���al. (PNAS, May 2025), with fresh clarity:

���� What���s the Big Idea?

Astrocytes

, star-shaped glial cells, aren���t just support���they

actively participate

in memory.

Each astrocyte reaches out with millions of

tendrils (processes)

that wrap around synapses, forming

tripartite synapses

with neuron pairs

arxiv.org

+10

sciencedaily.com

+10

technologynetworks.com

+10

arxiv.org

+6

bioengineer.org

+6

rfsafe.com

+6

.

When neurons fire, astrocyte processes detect neurotransmitters and respond via

calcium (Ca�����) signaling

, which can modulate synaptic strength with gliotransmitters���completing a feedback loop

rfsafe.com

+2

bioengineer.org

+2

rfsafe.com

+2

.

���� How It Becomes a Memory System

1.

Tripartite Synapses + Calcium Waves

Think of each synapse as having a ���listener��� (astrocyte process) that picks up signals, integrates them, and influences future neural activity���like a smart ink that adjusts how hard neurons ���write��� their connections.

2.

Dense, Multi-Neuron Coupling

Astrocytes

link many synapses

together���not just pairs���effectively enabling

higher-order interactions

(e.g., three or four neurons influencing each other simultaneously)

.

This is akin to how

Dense Associative Memory (DAM)

networks work: they can recall patterns with partial cues and have high capacity, due to rich interaction patterns.

3.

Energy-Based Attractors

The system is built on a

global energy function

derived from neural, synaptic, and astrocytic Lagrangians.

All activity dynamics (neuronal firing, synaptic change, calcium waves) aim to

reduce this energy

, naturally settling into stable ���memory attractor��� states

technologynetworks.com

+12

pnas.org

+12

arxiv.org

+12

rfsafe.com

.

4.

Superior Memory Scaling

In standard Hopfield models, memory capacity grows

linearly

with neuron count

N

N

N

.

The astrocyte-augmented system achieves

supralinear scaling

(~

N

2

N^2

N

2

memory units) because

each astrocyte process acts as an additional memory component

, interconnecting synapses densely

technologynetworks.com

+4

arxiv.org

+4

arxiv.org

+4

sciencedaily.com

+11

bioengineer.org

+11

arxiv.org

+11

.

���� Why Astrocytes Matter

Because astrocytes link many synapses,

memories can be stored in their Ca����� dynamics

across processes���not just in synaptic weights.

This enables the brain to store an

extraordinarily large number of memories

, far outpacing neuron-only storage models

arxiv.org

+3

rfsafe.com

+3

bioengineer.org

+3

scienceblog.com

+7

washingtonpost.com

+7

rfsafe.com

+7

.

Astrocytes thus act as a

memory ���cloud���

, enhancing capacity and robustness.

���� Step-by-Step in Plain Terms

Neurons fire

��� neurotransmitter release at a synapse.

Astrocyte process detects it

��� Ca����� level rises in that tendril.

Astrocyte internally diffuses Ca����� across processes ���

linking multiple synapses

.

It then releases gliotransmitters ���

modulates synaptic strength

, changing how new spikes affect neurons.

This loop is continuous, creating

stable patterns of activity

corresponding to stored memories.

���� Think Of It Like This

Neurons

are like

letters

in a word.

Synapses

are like

bigrams

���how letters pair.

Astrocytes + Ca�����

act like a

word-level editor

, not just dealing with pairs but whole contexts.

They build

bigger patterns

���like sentences or paragraphs���allowing

richer memory structures

.

���� Testable Prediction

Block astrocytic Ca����� diffusion (e.g., pharmacologically) ��� recall and memory capacity should drop significantly. That's a direct test of the model

arxiv.org

+10

bioengineer.org

+10

rfsafe.com

+10

arxiv.org

technologynetworks.com

+1

arxiv.org

+1

pubmed.ncbi.nlm.nih.gov

.

���� Why It Matters

Reveals

new biological hardware

for memory.

Bridges neuroscience with

modern AI architectures

(DAM and Transformer mechanisms).

Suggests future

AI and neuromorphic designs

might leverage ���astrocyte-like��� components for superior memory.

Would you like a diagram tracing this loop, or a mathematical overview linking its components to Dense Associative Memory and Transformers?

Sources

Tools

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