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The Hydrodynamic Limit of Neural Networks with Balanced Excitation and Inhibition

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Balanced neuron networks converge to four exact equations

desk verdict A novel and well-motivated model for balanced E/I networks with synaptic noise, but the main theorem is not proven: the key dissipativity lemma is false as stated and the covariance convergence is missing. read the letter →

arxiv 2412.17273 v1 pith:BC3ZYTB6 submitted 2024-12-23 math-ph math.MPmath.PRq-bio.NC

James MacLaurin , Pedro Vilanova This is my paper · ORCID
classification math-phmath.MPmath.PRq-bio.NC MSC 60K3560F0560G5592B20
keywords balancedneuralnetworkshydrodynamiclimitexcitation-inhibitionbalancePoissonspikingGaussianfluctuationsmean-fieldPiecewiseDeterministicMarkovProcesslargedeviations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish a hydrodynamic limit for a fully connected network of $n$ excitatory and $n$ inhibitory neurons whose interactions scale as $n^{-1/2}$, much stronger than the usual $n^{-1}$ mean-field scaling. The authors prove that, as $n \to \infty$ and for every time up to the moment the system leaves the balanced manifold, the empirical distribution of synaptic activities becomes Gaussian and its mean and variance obey a closed system of four ordinary differential equations. A sympathetic reader should care because this turns a popular verbal explanation for cortical variability -- the dynamic balance of excitation and inhibition -- into a precise limit theorem with explicit equations.

What carries the argument

The object carrying the argument is the balanced manifold $U$, the set of mean-covariance states $(v_e, v_i, K_e, K_i)$ at which the mean excitatory and inhibitory input fields $F_e, F_i$ vanish and the Jacobian $J_v$ of these fields has eigenvalues with negative real parts. The load-bearing step is the dissipativity estimate of Lemma 5.7, $q^T J q \le -\zeta_t |q|^2$, which shows that deviations of the system-wide mean from the limit are exponentially damped as long as the empirical law is near the Gaussian limit. This damping is what lets the $O(n^{-1/2})$ synaptic noise remain harmless, allowing the variance equations to close and the empirical measure to become Gaussian.

What would settle it

Compute the real parts of the eigenvalues of the Jacobian $J_v$ along the ODE trajectory of Lemma 4.2; if any becomes non-negative at a time $t < \eta$, the balanced manifold is not attracting. Simulate the $2n$-neuron system in that parameter regime: if the empirical means and covariances visibly diverge from the ODE solution before time $\eta$, the hydrodynamic limit as stated does not hold there.

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Extended reading notes

Core claim

The central claim is Theorem 4.1: with unit probability, for any $T < \eta$, the empirical means $\hat{v}^n_e, \hat{v}^n_i$ and covariances $\hat{K}^n_e, \hat{K}^n_i$ converge uniformly on $[0,T]$ to the solution $(\bar{v}_e, \bar{v}_i, K_e, K_i)$ of the autonomous ODE system (4.6)-(4.13) constrained to the balanced manifold $U$. Along the way the empirical measure of the $2n$ synaptic variables is shown to concentrate on a Gaussian law: the means are pinned by the balance conditions $F_e = F_i = 0$, while the variances evolve through an Ornstein-Uhlenbeck-type equation driven by the limiting firing rates. The limit holds only while the balanced manifold remains attracting; at the exit time $\eta$ the ODE system leaves $U$ and the theorem no longer applies.

Load-bearing premise

The load-bearing premise is that the balanced state keeps pulling the network back toward it for the whole time interval: if that restoring influence ever vanishes or reverses before the time horizon, the proof of the hydrodynamic limit collapses, and the paper only checks this numerically for its examples.

Editorial extensions

If this is right

  • The population-level mean activity $\hat{v}^n_e, \hat{v}^n_i$ concentrates on the ODE mean $\bar{v}_e, \bar{v}_i$ uniformly up to time $T < \eta$, so macroscopic activity is deterministic in the infinite-size limit.
  • The empirical covariances converge to $K_e, K_i$, so trial-to-trial variability at the population level is described by just two variance equations and not by $2n$ coupled random trajectories.
  • The limiting law is Gaussian, meaning the balanced state produces exactly the irregular, asynchronous fluctuation picture that the balanced-network theory was created to explain.
  • For $t \ge \eta$ the theorem gives no prediction; the authors conjecture an abrupt, discontinuous change in activity when balance breaks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An unstated consequence of the Gaussian limit is that the empirical distribution's skewness and higher cumulants should vanish as $n$ grows; one could test this directly by measuring the third and fourth empirical moments in the same simulations.
  • Because the stability condition is checked numerically rather than proved, a natural extension is to map out which parameter regions make $\zeta_t$ positive before $\eta$; those regions should show the predicted breakdown of the Gaussian description.
  • The same balance-damping mechanism could plausibly carry over to networks with sparse random connectivity, where the effective interaction strength per neuron would need to be rescaled; the paper's all-to-all proof would need a new argument for the mean-field approximation of the firing-rate sums.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper studies an all-to-all network of n excitatory and n inhibitory neurons whose synaptic variables follow linear ODEs and receive Poisson spike trains with sigmoidal rates, with interactions scaled as n^{-1/2}. The authors define a 'balanced manifold' U by cancellation of mean excitation and inhibition together with local stability of the mean-field Jacobian, and state that if the initial empirical measures converge to Gaussian measures on U, then for times before the manifold is left, the empirical mean and variance converge almost surely to the solution of a finite ODE system (4.6)-(4.13). The proof strategy is to decompose each trajectory into a system-wide mean and fluctuations, prove Wasserstein-1 convergence of the empirical fluctuations to a Gaussian process via KMT coupling and Sanov's theorem, and then use the attracting nature of U to damp the mean fluctuations.

Significance. If the main theorem were fully proved, the contribution would be significant: it gives a rigorous hydrodynamic limit for a balanced network with O(n^2) Poisson noise sources and n^{-1/2} coupling, a regime distinct from typical McKean-Vlasov/Hawkes mean-field limits, and it yields closed ODEs for the mean and variance. The paper also provides numerical simulations supporting the limit. However, the current manuscript contains a central invalid estimate in Lemma 5.7, an incomplete proof of Theorem 4.1, and a Borel-Cantelli argument with a non-summable tail, so the significance is conditional on substantial revision.

major comments (4)
  1. [Section 5, Lemma 5.7 (Eqs. (5.80)-(5.81))] The key dissipativity estimate is not established. In the proof of Lemma 5.7, after Eq. (5.80), the authors write that the quadratic form q^T J(bar v, mu) q is at most -zeta_t |q|^2 because 'the balanced manifold is by definition attracting', and then extend this to J(tilde v). However, Hypothesis (3.8) only constrains the real parts of the eigenvalues of J_v, not the symmetric part of J_v. For a non-symmetric 2x2 matrix, q^T J q is governed by (J + J^T)/2, which can have a positive eigenvalue while all eigenvalues of J have negative real parts; for example, J = [[1,-2],[2,-3]] has eigenvalues -1 and -1 but q^T J q = 1 for q = (1,0). In the present model with increasing sigmoidal f's, partial_{v_e} F_e >= 0 and partial_{v_i} F_i <= 0, so the quadratic form can indeed be positive for q = (1,0). Consequently Lemma 5.4 and the damping of the O(n^{-1/2}) mean fluctuations are not proved. The authors must either prove a uniform bound q^T J_v q <= -zeta |q|^2 from the model parameters or replace this step with a Lyapunov-function or weighted-norm argument.
  2. [Section 5, Proof of Theorem 4.1 (after Corollary 5.3)] The proof of the main theorem is incomplete. After Corollary 5.3, the text states that it suffices to show (5.20)-(5.23) and then stops; no argument is given for (5.20)-(5.21). The preceding lemmas only provide Wasserstein-1 estimates (Lemmas 5.1, 5.2, 5.4). Wasserstein-1 convergence of empirical measures to a Gaussian does not control the empirical second moment, so the convergence of hat K^n_e and hat K^n_i to K_e and K_i does not follow. A separate estimate for |hat K^n_alpha(t) - K_alpha(t)|, for example via uniform integrability or an L^2 or quadratic Wasserstein bound, is required.
  3. [Section 5, Lemma 5.2, Eq. (5.30)] The Borel-Cantelli step in the proof of Lemma 5.2 is invalid as written. The tail bound P(X^c_{epsilon,n}) <= exp(-C_epsilon n^{-1/2}) in (5.30) is not summable in n, so it cannot imply that X_{epsilon,n} holds almost surely for all large n. The standard KMT strong approximation for Poisson processes gives a much stronger, summable tail (typically exp(-c epsilon sqrt n) for fixed epsilon), so this may be a typographical error in the exponent, but the rate must be corrected because Corollary 5.3 and the 'with unit probability' statement in Theorem 4.1 depend on this step.
  4. [Section 5, Lemma 5.6, Eq. (5.69)] The proof of Lemma 5.6 contains a circular dependence. In (5.69), the difference between Q^n_{alpha beta}(t) and the Gaussian integral is bounded by c |bar v_beta(t) - v^n_beta(t)| + c d_W(hat mu^n_{beta,t}, nu^n_{beta,t}) + c |K^n_{alpha beta}(t) - K_{alpha beta}(t)|, and the lemma is then declared to follow. Since K^n_{alpha beta}(t) is the empirical covariance whose convergence is part of the desired conclusion, this is circular unless an independent estimate for |K^n_{alpha beta}(t) - K_{alpha beta}(t)| is supplied. This gap is connected to the previous comment and also affects the proof of Lemma 5.1.
minor comments (5)
  1. [Section 3, Assumptions] The text says 'in fact we do this in Section ' with a blank section number; the intended reference is missing.
  2. [Section 5, Lemma 5.5] The statement of Lemma 5.5 contains the typo 'na^{-1} log P'; it should read 'n^{-1} log P'.
  3. [Section 5, Eq. (5.38)] In the definition tilde u^j_{alpha,t} = tilde x^j_{alpha,t} + v_{alpha,t}, the symbol v_{alpha,t} is not defined at that point; it should presumably be v^n_{alpha}(t).
  4. [Section 5, proof of Lemma 5.7] In the intermediate value step, the text writes tilde v_e = a bar v_e(t) + (1-a) v^n_e(t) with a single parameter a for both components; since the two components may require different convex parameters, the notation should be clarified.
  5. [Section 6, Numerical Simulations] The captions for the six panels are missing from the text, so the reader cannot tell which panel corresponds to K^n_i versus K^n_e or v^n_i versus v^n_e; please add figure captions and axis labels.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the hydrodynamic-limit ODEs are derived from the microscopic dynamics rather than fitted, and the self-citations are background only.

full rationale

The derivation chain is not circular in the sense defined here. The limiting ODEs (4.6)-(4.13) are obtained from the microscopic equations: the variance equations come from the Ornstein-Uhlenbeck-type covariance computations in Lemma 5.6 (eqs. (5.63)-(5.64)), and the mean equations are obtained by differentiating the balance conditions Fe=Fi=0 using the implicit function theorem in the proof of Lemma 4.2 (eq. (5.24)). These quantities are not fitted to simulation output. In the numerical section, the authors solve Fe=Fi=0 only to place the initial means on the balanced manifold as required by Hypothesis 3.2, and then compare the stochastic simulation to the independently integrated ODE system. Hypothesis 3.2's requirement that the initial condition lie in U is an assumption on the initial state, not a re-labeled conclusion. The self-citations ([1], [8], [9], [16], [29], [40]) are background, motivation, or comparison references; none supplies the load-bearing stability step or a uniqueness theorem on which the proof rests. The closest point to a definitional shortcut is Lemma 5.7, which writes "since the balanced manifold is by definition attracting" to justify q^T J q <= -zeta |q|^2 from the eigenvalue condition in (3.8); for a non-symmetric J this is a mathematical gap, not a circular reduction of the theorem to its inputs. Accordingly, no circular step meets the quote-and-reduction bar, and the paper is self-contained against its stated assumptions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on smoothness and boundedness of the firing-rate functions, convergence of initial conditions to the balanced manifold, and local stability of that manifold. No free parameters are fitted to data in the theorem; the simulation constants are illustrative. No new physical entities are introduced.

assumptions (4)
  • domain assumption Hypothesis 3.1: the intensity functions f_alpha_beta are globally Lipschitz and satisfy 0 < f_alpha_beta <= C_f.
    Used throughout to apply time-rescaled Poisson representations and KMT approximation; sigmoidal firing rates satisfy this, but the paper does not prove blow-up without it.
  • domain assumption Hypothesis 3.2: initial empirical measures converge to Gaussian measures with means and variances on the balanced manifold U.
    The theorem is proved only for initial conditions concentrating on the balanced manifold; this is stated but not relaxed or verified from first principles.
  • domain assumption The balanced manifold U is locally attracting: the Jacobian J_v has eigenvalues with strictly negative real parts (definition (3.8)).
    Lemma 5.7 uses this to get the dissipativity bound q^T J q <= -zeta |q|^2; without it, the system may leave the manifold and the limit is not proved.
  • standard math Standard probabilistic and analytic theorems: KMT approximation, Sanov's theorem, Gronwall's inequality, and the implicit function theorem.
    Invoked in the proofs without derivation; these are standard background results.

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Pith. "Pith review of The Hydrodynamic Limit of Neural Networks with Balanced Excitation and Inhibition." pith.science (2026). https://pith.science/paper/BC3ZYTB6

@misc{pith2026241217273,
  author       = {Pith},
  title        = {Pith review of: The Hydrodynamic Limit of Neural Networks with Balanced Excitation and Inhibition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BC3ZYTB6}},
  note         = {Machine review of arXiv:2412.17273}
}
read the original abstract

The theory of `Balanced Neural Networks' is a very popular explanation for the high degree of variability and stochasticity in the brain's activity. We determine equations for the hydrodynamic limit of a balanced all-to-all network of 2n neurons for asymptotically large n. The neurons are divided into two classes (excitatory and inhibitory). Each excitatory neuron excites every other neuron, and each inhibitory neuron inhibits all of the other neurons. The model is of a stochastic hybrid nature, such that the synaptic response of each neuron is governed by an ordinary differential equation. The effect of neuron j on neuron k is dictated by a spiking Poisson Process, with intensity given by a sigmoidal function of the synaptic potentiation of neuron j. The interactions are scaled by n^{-1/2} , which is much stronger than the n^{-1} scaling of classical interacting particle systems. We demonstrate that, under suitable conditions, the system does not blow up as n asymptotes to infinity because the network activity is balanced between excitatory and inhibitory inputs. The limiting population dynamics is proved to be Gaussian: with the mean determined by the balanced between excitation and inhibition, and the variance determined by the Central Limit Theorem for inhomogeneous Poisson Processes. The limiting equations can thus be expressed as autonomous Ordinary Differential Equations for the means and variances.

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Reference graph

Works this paper leans on

58 extracted references · 16 canonical work pages

  1. [1]

    Adams and James MacLaurin

    Zachary P. Adams and James MacLaurin. The isochronal phase of stochastic pde and integral equations: Metastability and other properties. Journal of Differential Equations , 414:773–816, 1 2025

    2025

  2. [2]

    Multivariate hawkes processes on inhomogeneous random graphs

    Zo´ e Agathe-Nerine. Multivariate hawkes processes on inhomogeneous random graphs. Stochastic Processes and their Applications , 152:86–148, 10 2022

    2022

  3. [3]

    Large deviations theory for markov jump models of chemical reaction networks

    Andrea Agazzi, Amir Dembo, and Jean Pierre Eckmann. Large deviations theory for markov jump models of chemical reaction networks. Annals of Applied Probability , 28:1821–1855, 2018

    2018

  4. [4]

    Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex

    Daniel Amit and Nicolas Brunel. Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. Cerebral Cortex, 7:237–252, 1997

    1997

  5. [5]

    Stochastic Analysis of Biochemical Systems

    David F Anderson and Thomas G Kurtz. Stochastic Analysis of Biochemical Systems . Springer, 2015

    2015

  6. [6]

    Large deviations for langevin spin glass dynamics

    G Ben Arous and A Guionnet. Large deviations for langevin spin glass dynamics. Probability Theory and Related Fields, 102, 1995

    1995

  7. [7]

    Cugliandolo-kurchan equations for dynamics of spin- glasses

    Gerard Ben Arous, Amir Dembo, and Alice Guionnet. Cugliandolo-kurchan equations for dynamics of spin- glasses. Probability Theory and Related Fields , 136:619–660, 2006

    2006

  8. [8]

    Large deviations of hawkes processes on structured sparse random graphs

    D Avitabile and J Maclaurin. Large deviations of hawkes processes on structured sparse random graphs. 2024

    work page 2024

Show all 58 references
  1. [9]

    Neural fields and noise-induced patterns in neurons on large disordered networks

    Daniele Avitabile and James Maclaurin. Neural fields and noise-induced patterns in neurons on large disordered networks. Arxiv 2408.12540v1, 2024

    2024 arXiv

  2. [10]

    Mean-field description and propa- gation of chaos in networks of hodgkin-huxley and fitzhugh-nagumo neurons

    Javier Baladron, Diego Fasoli, Olivier Faugeras, and Jonathan Touboul. Mean-field description and propa- gation of chaos in networks of hodgkin-huxley and fitzhugh-nagumo neurons. Journal of Mathematical Neuroscience, 2:1–67, 2012

    2012

  3. [11]

    Noise-Induced Phenomena in Slow-Fast Dynamical Systems

    Nils Berglund and Barbara Gentz. Noise-Induced Phenomena in Slow-Fast Dynamical Systems . Springer, 2006

    2006

  4. [12]

    Convergence of Probability Measures

    Patrick Billingsley. Convergence of Probability Measures. 1999

    1999

  5. [13]

    Pattern formation in random networks using graphons

    Jason Bramburger and Matt Holzer. Pattern formation in random networks using graphons. SIAM Journal on Mathematical Analysis , 55:2150–2185, 2023

    2023

  6. [14]

    Probability Theory and Stochastic Processes

    Pierre Bremaud. Probability Theory and Stochastic Processes. Springer, 2020

    2020

  7. [15]

    Stochastic Processes in Cell Biology

    Paul Bressloff. Stochastic Processes in Cell Biology. I . Springer, 2 edition, 2021

    2021

  8. [16]

    Bressloff and J.N

    P.C. Bressloff and J.N. Maclaurin. Stochastic hybrid systems in cellular neuroscience. Journal of Mathematical Neuroscience, 8, 2018

    2018

  9. [17]

    Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons

    Nicolas Brunel. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience , 8:183–208, 2000

    2000

  10. [18]

    Effects of neuromodulation in a cortical network model of object working memory dominated by recurrent inhibition, 2001

    Nicolas Brunel and Xiao-Jing Wang. Effects of neuromodulation in a cortical network model of object working memory dominated by recurrent inhibition, 2001

    2001

  11. [19]

    Propagation of chaos: a review of models, methods and applications

    Louis-Pierre Chaintron and Antoine Diez. Propagation of chaos: a review of models, methods and applications. i. models and methods. Kinetic and Related Models , 15, 2 2022

    2022

  12. [20]

    Spin Glass Theory and Far Beyond

    Patrick Charbonneau, Enzo Marinari, Mark Mezard, Giorgio Parisi, Federico Ricci-Tersenghi, Gabriella Sicuro, and Francesco Zamponi, editors. Spin Glass Theory and Far Beyond . World Scientific, 2023

    2023

  13. [21]

    Mean-field limit of generalized hawkes processes

    Julien Chevallier. Mean-field limit of generalized hawkes processes. Stochastic Processes and their Applications, 127:3870–3912, 2017

    2017

  14. [22]

    Crisanti and H

    A. Crisanti and H. Sompolinsky. Path integral approach to random neural networks. Physical Review E , 98:1–20, 2018

    2018

  15. [23]

    Andrea Crisanti, Heinz Horner, and H.J. Sommers. The spherical p-spin interaction spin-glass model. Zeit- schrift fur Physik B Condensed Matter , 92:257–271, 1993

    1993

  16. [24]

    Large Deviations Techniques and Applications 2nd Edition

    Amir Dembo and Ofer Zeitouni. Large Deviations Techniques and Applications 2nd Edition . Springer, 1998

    1998

  17. [25]

    Interacting hawkes processes with multiplicative inhibition, 2022

    C´ eline Duval, Eric Lu¸ con, and Christophe Pouzat. Interacting hawkes processes with multiplicative inhibition, 2022

    2022

  18. [26]

    Rainer Engelken, Alessandro Ingrosso, Ramin Khajeh, and Sven Goedeke L.F. Abbott. Input correlations impede suppression of chaos and learning in balanced firing-rate networks. PLoS Computational Biology, 18, 12 2022

    2022

  19. [27]

    Bard Ermentrout and David H

    G. Bard Ermentrout and David H. Terman. Mathematical Foundations of Neuroscience . Springer, 2010

    2010

  20. [28]

    Mean field limits for interacting hawkes processes in a diffusive regime

    Xavier Erny, Eva L¨ ocherbach, and Dasha Loukianova. Mean field limits for interacting hawkes processes in a diffusive regime. Bernoulli, 28:125–149, 2 2021

    2021

  21. [29]

    Asymptotic description of neural networks with correlated synaptic weights

    Olivier Faugeras and James MacLaurin. Asymptotic description of neural networks with correlated synaptic weights. Entropy, 17:4701–4743, 2015

    2015

  22. [30]

    Statistical field theory for neural networks

    Moritz Helias and David Dahmen. Statistical field theory for neural networks. pages 1–107, 2019

    2019

  23. [31]

    Mean field limit for stochastic particle systems

    Pierre-Emmanuel Jabin and Zhenfu Wang. Mean field limit for stochastic particle systems. Active Particles, Volume 1: Advances in Theory, Models, and Applications , 2017. 17

    2017

  24. [32]

    Brownian Motion and Stochastic Calculus 2nd Edition

    Ioannis Karatzas and Steven Shreve. Brownian Motion and Stochastic Calculus 2nd Edition . Springer, 1991

    1991

  25. [33]

    Solutions of a stochastic differential equation forced onto a manifold by a large drift

    G.S Katzenberger. Solutions of a stochastic differential equation forced onto a manifold by a large drift. The Annals of Probability, 19:1587–1628, 1991

    1991

  26. [34]

    An approximation of partial sums of independent rv’-s, and the sample df

    J Komlds, P Major, and G Tusnfidy. An approximation of partial sums of independent rv’-s, and the sample df. i, 1975

    1975

  27. [35]

    Koml´ os, P

    J. Koml´ os, P. Major, and G. Tusn´ ady. An approximation of partial sums of independent rv’s, and the sample df. ii. Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 34:33–58, 3 1976

    1976

  28. [36]

    Sukbin Lim and Mark S. Goldman. Balanced cortical microcircuitry for spatial working memory based on corrective feedback control. Journal of Neuroscience , 34:6790–6806, 2014

    2014

  29. [37]

    Slow dynamics and high variability in balanced cortical networks with clustered connections

    Ashok Litwin-Kumar and Brent Doiron. Slow dynamics and high variability in balanced cortical networks with clustered connections. Nature Neuroscience, 15:1498–1505, 2012

    2012

  30. [38]

    Spiking neurons: Interacting hawkes processes, mean field limits and oscillations

    E Locherbach. Spiking neurons: Interacting hawkes processes, mean field limits and oscillations. ESAIM: Proceedings and Surveys, 60:90–103, 2018

    2018

  31. [39]

    Quenched asymptotics for interacting diffusions on inhomogeneous random graphs

    Eric Lucon. Quenched asymptotics for interacting diffusions on inhomogeneous random graphs. Stochastic Processes and their Applications , 130:6783–6842, 11 2020

    2020

  32. [40]

    Population-level activity in large random neural networks

    James MacLaurin, Moshe Silverstein, and Pedro Vilanova. Population-level activity in large random neural networks. Under Review in Annals of Applied Probability , 2024

    2024

  33. [41]

    Detecting and estimating signals in noisy cable structures, i: Neuronal noise sources

    Amit Manwani and Christof Koch. Detecting and estimating signals in noisy cable structures, i: Neuronal noise sources. Neural Computation, 11:1797–1829, 1999

    1999

  34. [42]

    De Masi, A

    A. De Masi, A. Galves, E. L¨ ocherbach, and E. Presutti. Hydrodynamic limit for interacting neurons. Journal of Statistical Physics , 158:866–902, 2014

    2014

  35. [43]

    Phase changes in neuronal postsynaptic spiking due to short term plasticity

    Mark D Mcdonnell and Bruce P Graham. Phase changes in neuronal postsynaptic spiking due to short term plasticity. PLOS Computational Biology , pages 1–26, 2017

    2017

  36. [44]

    Large deviations and mean-field theory for asymmetric random recurrent neural networks

    Olivier Moynot and Manuel Samuelides. Large deviations and mean-field theory for asymmetric random recurrent neural networks. Probability Theory and Related Fields , 123:41–75, 5 2002

    2002

  37. [45]

    Parsons and Tim Rogers

    Todd L. Parsons and Tim Rogers. Dimension reduction for stochastic dynamical systems forced onto a manifold by large drift: A constructive approach with examples from theoretical biology. Journal of Physics A: Mathematical and Theoretical, 50, 2017

    2017

  38. [46]

    Pfaffelhuber, S

    P. Pfaffelhuber, S. Rotter, and J. Stiefel. Mean-field limits for non-linear hawkes processes with excitation and inhibition. Stochastic Processes and their Applications , 153:57–78, 11 2022

    2022

  39. [47]

    Balanced networks of spiking neurons with spatially dependent recurrent connections

    Robert Rosenbaum and Brent Doiron. Balanced networks of spiking neurons with spatially dependent recurrent connections. Physical Review X , pages 1–9, 2014

    2014

  40. [48]

    Smith, Adam Kohn, Jonathan E

    Robert Rosenbaum, Matthew A. Smith, Adam Kohn, Jonathan E. Rubin, and Brent Doiron. The spatial structure of correlated neuronal variability. Nature Neuroscience, 20:107–114, 2017

    2017

  41. [49]

    Ran Rubin, L. F. Abbott, and Haim Sompolinsky. Balanced excitation and inhibition are required for high- capacity, noise-robust neuronal selectivity. Proceedings of the National Academy of Sciences of the United States of America , 114:E9366–E9375, 2017

    2017

  42. [50]

    Noise, neural codes and cortical organization

    Michael N Shadlen and William T Newsome. Noise, neural codes and cortical organization. Current Opinion in Neurobiology, 4:569–579, 1994

    1994

  43. [51]

    The variable discharge of cortical neurons : Implications for connectivity , computation , and information coding

    Michael N Shadlen and William T Newsome. The variable discharge of cortical neurons : Implications for connectivity , computation , and information coding. The Journal of Neuroscience , 18:3870–3896, 1998

    1998

  44. [52]

    The highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps

    William Softky and Christof Koch. The highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps. The Journal of Neuroscience , 13:334–350, 1993

    1993

  45. [53]

    Time-dependent order parameters in spin-glasses

    H Sompolinsky. Time-dependent order parameters in spin-glasses. Physical Review Letters, 47:935–938, 1981

    1981

  46. [54]

    Topics in Propagation of Chaos

    Alain-Sol Sznitman. Topics in Propagation of Chaos . Springer-Verlag, 1989

    1989

  47. [55]

    Bistability and up / down state alternations in inhibition- dominated randomly connected networks of lif neurons

    Elisa M Tartaglia and Nicolas Brunel. Bistability and up / down state alternations in inhibition- dominated randomly connected networks of lif neurons. Scientific Reports, pages 1–14, 2017

    2017

  48. [56]

    M. V. Tsodyks and T. Sejnowski. Rapid state switching in balanced cortical network models. Network: Computation in Neural Systems , 6:111–124, 1995

    1995

  49. [57]

    Chaos in neuronal networks with balanced excitatory and inhibitory activity

    C Van Vreeswijk and H Sompolinsky. Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science, 274:1724–1726, 1996

    1996

  50. [58]

    Van Vreeswijk and H

    C. Van Vreeswijk and H. Sompolinsky. Chaotic balanced state in a model of cortical circuits. Neural Compu- tation, 10:1321–1371, 1998. 18

    1998

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