Balanced Dynamics in Strongly Coupled Networks

TL;DR

The paper investigates how strongly coupled stochastic networks can automatically reach a balanced excitation–inhibition regime, formulating a general conjecture that a balance manifold $\mathcal{B}$ governs asymptotic dynamics under diverging coupling. It frames the problem as a double-limit in the space of probability measures and develops two complementary viewpoints: a fast, short-time balance picture (Method M2) and a mean-field, desingularization approach (Method M1) leading to a Hamilton–Jacobi/Hopf–Cole analysis. Applying these ideas to FitzHugh–Nagumo networks with electrical and chemical coupling, the authors prove a complete asymptotic result in a one-dimensional separable-coupling model: under $n\to\infty$ then $\gamma\to\infty$, solutions concentrate on the balance manifold, with balance voltages determined by explicit formulas. The findings offer a minimalistic explanation for cortex-wide balance and open avenues for cross-disciplinary applications where strong coupling could induce rapid convergence to balanced states.

Abstract

Many mathematical models of interacting agents assume that individual interactions scale down in proportion to the network size, ensuring that the combined input received from the network does not diverge. In theoretical neuroscience, Sompolinsky and Van Vreeswijk proposed in 1996 that, should these scalings be violated (and under appropriate conditions), the system may not diverge but rather approach a balanced state where the inputs to each neuron compensate each other (in neuroscience, where inhibitory currents compensate the excitatory ones). We come back to this observation and formulate here a mathematical conjecture for the occurrence of such behaviors in general stochastic systems of interacting agents. From a mathematical viewpoint, this conjecture can be viewed as a double-limit problem in the space of probability measures, which we discuss in detail, as it provides several possible mathematical avenues for proving this result. We provide some numerical and theoretical explorations of the conjecture in classical models of neuronal networks. Moreover, we provide a complete proof of an asymptotic result consistent with one of the double-limit problems in a one-dimensional model with separable coupling inspired by models of chemically-coupled neurons. This proof relies on asymptotic methods, and particularly desingularization techniques used in some PDEs, that we apply here to the mean-field limit of the network as the coupling is made to diverge. From the applications viewpoint, this theory provides an alternative, minimalistic explanation for the widely observed balance of excitation and inhibition in the cerebral cortex not requiring the assumption of the existence of complex regulatory mechanisms.

Figures (3)

• Figure 1: Dynamics of the electrically coupled FitzHugh Nagumo with $n=300$ neurons with $\gamma(n)=n$ (A,B,C,D) or $\gamma(n)=\sqrt{n}$ (A',B',C',D'). (A,A') voltage variable for 20 neurons in the network (with inset showing the rapid convergence to balance), (B,B') associated recovery variable, (C,C') represent the distribution of the voltage (in logarithmic scale) at small timescales and (D,D') the standard deviation of both variables as a function of time. Quick convergence to the balanced regime is visible in the collapse of the distribution and of the standard deviation in the voltage variable. Simulations were performed using a custom Euler-Maruyama code on Matlab. $f(v)=v(1-v)(v-4)+4$, $a=0.005$, $b= 6$, $g=1$ and $\sigma=1$, and independent Gaussian initial condition with standard deviation equal to $5$, mean initial voltage $1$ and mean initial recovery variable $1.5$.
• Figure 1: Double-limit perspective on the balance conjecture. The original setup of the problem can be viewed as a specific path within the double-limit problem, where $n \ to \ infty$ and $\ gamma \ to \ infty$ separately, as shown in equation \ref{['eq:DoubleLim']}. The limit $n\to \infty$ with $\gamma$ fixed (step M1a) typically yields, under appropriate assumptions, McKean-Vlasov mean-field equations that reflect averaging effects (orange circle). The limit $\gamma\to\infty$ with $n$ fixed rather corresponds to a singular value problem, whereby a change of time shall provide us with a system of ODEs informing us about early behaviors (arising, when $\gamma$ is large but finite, at a timescale of $\gamma^{-1}$, pink circle and step M2a). The problem discussed in conjecture \ref{['conjecture']} is represented by the blue circle, and can be seen as a slanted path (orange arrow) in this double-limit graph, which we conjecture will collapse onto the balanced manifold $\mathcal{B}$. The rigorous results demonstrated in the paper follow the solid arrows (and hold under additional assumptions).
• Figure 2: Dynamics of the chemically coupled Fitzhugh Nagumo model \ref{['eq:EIFhN']}. (A1-2) Inhibition-dominated case with a stable balanced state ($g_{EE}=0.3$, $g_{EI}=2$, $g_{IE}=1$ and $g_{II}=10$), and (B1-2) Excitation-dominated regimes, where we see no convergence to a balanced state ($g_{EE}=1$, $g_{EI}=2$, $g_{IE}=0.1$ and $g_{II}=0.7$). (A2) and (B2) represent the dynamics of the system at short timescales for 20 excitatory (blue) and 20 inhibitory (red) neurons, together with the predicted value for the balanced excitatory (dashed black) and inhibitory (dashed purple) voltage. A clear, rapid convergence to the balanced state occurs in (A2), whereas in (B2), the voltages of neurons in each population escape the balanced regimes. Instead, they cluster into two groups: those converging to a common value above the balanced voltage and the rest converging to a common value below the balanced voltage. At longer timescales, those regimes remain stable, as shown in (A1) and (B1). (A1, greyed region) solutions quickly adjust to a 50% increase in excitatory conductances. Simulations were performed using a custom Euler-Maruyama code on Matlab, with independent, Gaussian initial conditions $\mathcal{N}(3,1)$ for the voltage and $\mathcal{N}(2,1)$ for the adaptation variable. Conductances uniformly distributed in $[0,2]$ (excitatory) or $[0,3]$ (inhibitory). Other parameters: $\gamma(N)=N/10$, with $f(x)=x(1-x)(x-0.3)$, $a=0.4$, $b=1.5$, $c=1$ and $\sigma=1$.

Theorems & Definitions (17)

• Conjecture 1
• Theorem 5.1
• Theorem 5.2
• Lemma A.1
• Proof 1
• Proof 2: Proof of Theorem \ref{['th:MR-existence']}
• Lemma A.2
• Proof 3
• Proposition B.1
• Proof 4