On the Dynamics of Random Neuronal Networks

Abstract

We study the mean-field limit and stationary distributions of a pulse-coupled network modeling the dynamics of a large neuronal assemblies. Our model takes into account explicitly the intrinsic randomness of firing times, contrasting with the classical integrate-and-fire model. The ergodicity properties of the Markov process associated to finite networks are investigated. We derive the large network size limit of the distribution of the state of a neuron, and characterize their invariant distributions as well as their stability properties. We show that the system undergoes transitions as a function of the averaged connectivity parameter, and can support trivial states (where the network activity dies out, which is also the unique stationary state of finite networks in some cases) and self-sustained activity when connectivity level is sufficiently large, both being possibly stable.

Appendix

1.1 Some Technical Results

Lemma 8

We assume that b is such that there exists $\gamma >0$ and $c>0$ such that

$$\begin{aligned} b'(x) \le \gamma b(x)+c \end{aligned}$$

(44)

Then, for any $\varepsilon >0$ and $p\in [1,3+\varepsilon ]$, there exist a constant $\gamma _1<(3+2\varepsilon )\gamma$, $c_1 >0$ and a value $\eta _b>0$ such for any $a\in (0,\eta _b)$ and $x\ge 0$,

$$\begin{aligned} b^p(x+a)-b^p(x) \le a \left( \gamma _1 b^p(x)+c_1 \right) . \end{aligned}$$

(45)

Proof

Let us start by noting that the inequality is trivial for b bounded. We will therefore assume in the rest of the proof that b diverges at infinity. We also remark that for any $1\le p<3+\varepsilon$, the map $b^p$ satisfies an inequality of type (44) where $\gamma$ is multiplied by p. Indeed, for any $\delta >0$, we can find $c_{\delta }>0$ such that:

$$\begin{aligned} \frac{\text {d} b^p(x)}{\text {d} x} \le p\gamma b^p(x) + p c b^{p-1}(x)\le (p\gamma +\delta ) b^p(x)+c_{\delta }. \end{aligned}$$

We will therefore demonstrate without loss of generality the proposition for $p=1$, and i.e. control the modulus of continuity of b under assumption (44). For an arbitrary $x_0>0$ and any $x\ge x_0$, we have:

$$\begin{aligned} \frac{b(x+a)}{b(x)} = \exp \left( \int _{x}^{x+a} \frac{b'(y)}{b(y)}\,\mathrm {d}y\right) \le e^{a \tilde{\gamma }}, \end{aligned}$$

with $\tilde{\gamma }=\gamma + {c}/{b(x_0)}$. We conclude that for $x\ge x_0$,

$$\begin{aligned} b(x+a)-b(x)\le (e^{a\tilde{\gamma }}-1)b(x). \end{aligned}$$

The map $a\mapsto (e^{a\tilde{\gamma }}-1)/a$ is smooth, non-decreasing and tends to $\tilde{\gamma }$ at $a=0$, which can be made arbitrarily close from $\gamma$ for sufficiently large $x_0$ (since b is unbounded). Therefore, there exists $x_0>0$ and $\eta >0$ such that for any $x\ge x_0$ and $a\in [0,\eta ]$,

$$\begin{aligned} b(x+a)-b(x)\le a (1+\varepsilon )\gamma b(x). \end{aligned}$$

Denoting $c_1$ the Lipschitz constant of b over the interval $[0,x_0+\eta ]$, we readily obtain (45) with $\gamma _1=\gamma (1+\varepsilon )$. $\square$

Another elementary property that is useful in our developments is the following:

Proposition 12

If x(t) is a non-negative càdlàg function on $\mathbb {R}_+$ and $\kappa>\delta >0$ such that, for A, $B\in \mathbb {R}$,

$$\begin{aligned} x(t)\le B+ x(s)-\int _s^t x(u)^\kappa \,\mathrm {d}u+ A\int _s^t x(u)^\delta \,\mathrm {d}u, \end{aligned}$$

holds for any $0\le s\le t$, then x(t) is uniformly bounded on any bounded time intervals.

Moreover, if (x(t)) is $C^1$ on $\mathbb {R}_+$ and $B=0$, we have a uniform bound for all times:

$$\begin{aligned} \sup _{t\ge 0} x(t)\le C_0<+\infty , \end{aligned}$$

where $C_0=x(0)\wedge A^{\kappa -\delta }$.

The proof is elementary once noted that the map $x\mapsto -x^\kappa + A x^\delta$ is upperbounded by a finite value $M>0$ and is strictly negative for any $x>A^{\kappa -\delta }$. The upperbound readily implies that $x(t)\le x_0+B+ M \,t$. For x continuously differentiable and $B=0$, the negativity of the integrand for $x>A^{\kappa -\delta }$ ensures that no trajectory exceeds $C_0=x(0) \wedge A^{\kappa -\delta }$.

1.2 Poisson Processes

The third elementary technical result used is related to the martingales associated to marked Poisson processes.

Proposition 13

If $\mathcal{N}$ is a Poisson process on $\mathbb {R}_+^3$ with intensity measure $\mathrm {d}u\otimes V(\mathrm {d}z)\otimes \mathrm {d}t$, f is a continuous function on $\mathbb {R}_+^3$ and $(Y(t)=(Y_1(t),Y_2(t))$ is a càdlàg adapted processes then the process (M(t)) defined by

$$\begin{aligned} \left( \int _{s=0}^t \int _{\mathbb {R}_+^2} f(Y(s-),z) \left[ \mathbbm {1}_{\{0\le u\le Y_1(s-)\}} \mathcal{N}(\mathrm {d}u,\mathrm {d}z,\mathrm {d}s)-Y_1(s)\,\mathrm {d}s\,V(\mathrm {d}z) \right] \right) \end{aligned}$$

is a local martingale whose previsible increasing process is given by

$$\begin{aligned} (\left\langle M\right\rangle (t))=\left( \int _0^t\mathrm {d}s\int _{\mathbb {R}_+} \mathrm {d}V(\mathrm {d}z) f(Y(s),z)^2\,Y_1(s)\right) . \end{aligned}$$

See Rogers and Williams [28] and Appendix B of Robert [27] for example.

1.3 Uniqueness

In the main text, we have shown tightness of the sequence of empirical measures, ensuring that the sequence is relatively compact. Moreover, we showed that the possible limits are time-dependent measures $\Lambda (t)$ that satisfy, for all $f\in C^{1}(\mathbb {R})$, Eq. (36) that we write here as:

$$\begin{aligned} \left\langle \Lambda (t),\, f\right\rangle= & {} \left\langle \Lambda (0),\, f\right\rangle +\int _0^t\left[ \left\langle \Lambda (u),\, -x f'(x) + E(V) \left\langle \Lambda (u),\, b\right\rangle f'(x)\right\rangle \right. \nonumber \\&\left. -\left\langle \Lambda (u),\, (f(x)-f(0))\,b(x)\right\rangle \right] \,\mathrm {d}u. \end{aligned}$$

(46)

In the above notations, x is a generic symbolic variable, which we use for simplicity of notations, with the convention $\left\langle \Lambda (t),\, f(x)\right\rangle = \left\langle \Lambda (t),\, f\right\rangle$. We know that the law of the solution of the mean-field McKean–Vlasov Eq. (4) satisfies this system. We aim at showing there is a unique positive Radon measure such that the nonlinear Eq. (46) holds. We first remark that the differential equation conserves the total mass $\left\langle \Lambda (t),\, 1\right\rangle =\left\langle \Lambda (0),\, 1\right\rangle$. We are therefore searching for $\Lambda$ a probability measure satisfying the nonlinear Eq. (46). The proof of uniqueness uses the following properties:

Lemma 9

For any initial probability measure $\Lambda (0)$ of $\mathbb {R}_+$ with bounded support, if $\Lambda (t)$ is a solution of Eq. (46), there exists C and K such that

• (1) $\displaystyle \sup _{t\ge 0} \left\langle \Lambda (t),\, b\right\rangle \le C < \infty$.

• (2) $\Lambda (t)$ has its support in [0, K] for all $t\ge 0$.

Proof

The proof of (i) is similar to the analogous property shown on the possible solutions of the McKean–Vlasov equation. Denoting $B(t)=\left\langle \Lambda (t),\, b\right\rangle$ and using the inequality $b'(x)< \gamma b(x)+c$, we have:

$$\begin{aligned} B(t)&\le B(0) + \int _0^t \left\langle \Lambda (u),\, E(V) B(u) b'(x)-b(x)(b(x)-b(0))\right\rangle \mathrm {d}u\\&\le B(0) + \int _0^t \gamma E(V) B(u)^2 +(c+b(0)) B(u) -\left\langle \Lambda (u),\, b(x)^2\right\rangle \mathrm {d}u\\&\le B(0) + \int _0^t (\gamma E(V)-1)B(u)^2 + (c+b(0))B(u)\mathrm {d}u \end{aligned}$$

and we conclude using Proposition 12.

We now prove that any solution $\Lambda (t)$ to Eq. (46) has a uniformly bounded support. Let us assume that the support of $\Lambda (0)$ is contained in the interval $[0,K_0]$ and pick f a continuously differentiable and non-decreasing function such that

$$\begin{aligned} {\left\{ \begin{array}{ll} f(x)=0 &{} x<K\\ f(x)>0 &{} x>K \end{array}\right. } \end{aligned}$$

with $K=\max (K_0, \;C\,E(V))$ with C an upperbound of $\sup _{t\ge 0} \left\langle \Lambda (t),\, b\right\rangle$. Applying Eq. (46) to f and using the fact that $\left\langle \Lambda (t),\, b\right\rangle <C$, $fb\ge 0$ and $f'\ge 0$, we obtain the inequality:

$$\begin{aligned} 0\le \left\langle \Lambda (t),\, f\right\rangle \le \int _0^t \left\langle \Lambda (u),\, (-x+E(V) C) f'(x)\right\rangle \,\mathrm {d}u \le 0, \end{aligned}$$

hence $\left\langle \Lambda (t),\, f\right\rangle =0$ for all $t\ge 0$, implying that the support of $\Lambda (t)$ is contained within the compact set [0, K]. $\square$

With these a priori estimates on $\Lambda$ in hand, we can now show the uniqueness of possible solutions to the mean-field equation. For two probability measures $\lambda _1$ and $\lambda _2$, we define the distance:

$$\begin{aligned} \Vert \lambda _1 - \lambda _2\Vert _{\mathcal {S}}=\sup \left\{ \left\langle \lambda _1-\lambda _2, \; f\right\rangle :f\in \mathcal {S}\right\} , \end{aligned}$$

with

$$\begin{aligned} \mathcal {S}=\Big \{f\in C^1(\mathbb {R}_+): \Vert f\Vert _{\infty } \vee \Vert f'\Vert _{\infty } \le 1\Big \} \end{aligned}$$

and note that the subset of functions of $\mathcal {S}$ with bounded support is dense in the set of continuous functions with bounded support.

Proposition 14

Let $\Lambda (0)$ be a probability measure with bounded support, then Eq. (46) has a unique solution with initial condition $\Lambda (0)$.

Proof

We show that $\Vert \Lambda _1(t)-\Lambda _2(t)\Vert _\mathcal{S} =0$ for all times. Indeed, for any $f\in \mathcal {S}$, we have, denoting $\Delta (t)=\Lambda _1(t)-\Lambda _2(t)$,

$$\begin{aligned} \left\langle \Delta (t),\, f\right\rangle {=} \int _0^t \left\langle \Delta (u),\, {-}xf'(x){-}(f(x){-}f(0))b(x) {+}\left\langle \Lambda _1(u){,}f'\right\rangle b {+}\left\langle \Lambda _2(u){,}b\right\rangle f'\right\rangle \,\mathrm {d}u \end{aligned}$$

and therefore using the fact that $\Delta$ has a support included in the compact [0, K], we have:

$$\begin{aligned} \vert \left\langle \Delta (t),\, f\right\rangle \vert \le \Gamma (f,K)\int _0^t \Vert \Delta (u)\Vert _{\mathcal {S}}\,\mathrm {d}u \end{aligned}$$

(47)

with

$$\begin{aligned} \Gamma (f,K)&=\sup _{x\le K, u\ge 0} \Big \vert {-}x f'(x) {-}(f(x){-}f(0))b(x) {+} \left\langle \Lambda _1(u){,}f'\right\rangle b(x) {+} \left\langle \Lambda _2(u){,}b\right\rangle f'(x)\Big \vert \\&\le \mathcal {K}:= K +C + 3 b(K) \end{aligned}$$

We therefore have for all $f\in \mathcal {S}$ the inequality:

$$\begin{aligned} \vert \left\langle \Delta (t),\, f\right\rangle \vert \le \mathcal {K} \int _0^t \Vert \Delta (u)\Vert _{\mathcal {S}}\,\mathrm {d}u, \end{aligned}$$

which is therefore also valid for the norm of $\Delta$:

$$\begin{aligned} \Vert \Delta (t)\Vert _{\mathcal {S}}\le \mathcal {K} \int _0^t \Vert \Delta (u)\Vert _{\mathcal {S}}\,\mathrm {d}u. \end{aligned}$$

We conclude, by immediate recursion, that:

$$\begin{aligned} \sup _{s\le t}\Vert \Delta (s)\Vert _{\mathcal {S}}\le \mathcal {K} \int _0^t \sup _{s\le u}\Vert \Delta (s)\Vert _{\mathcal {S}}\,\mathrm {d}u \le \frac{\mathcal {K}^nt^n}{n!} \sup _{s\le t} \Vert \Delta (s)\Vert _{\mathcal {S}}, \end{aligned}$$

hence $\Vert \Delta (t)\Vert _{\mathcal {S}}=0$ for all $t\ge 0$. $\square$

1.4 Simulation Algorithms

This appendix describes the simulation algorithms used to obtain our plots for linear and quadratic rate functions in Sect. 7. We used two distinct algorithms: an exact simulation algorithm for the simulation of the extinction time, and for the sake of computational efficiency an approximate algorithm for large networks.

The algorithm we used in order to perform efficient simulations for large networks implements the evolution of the process at discrete times $t_k=k \delta t$ with $\delta t$ a small time step. In each time interval, we compute the probability that a spike occurs within the interval. We then draw a Bernoulli random variable with this probability, and update the network state accordingly.

This approximate dynamics allows to perform fast simulations and therefore to reach very large network size. However, computing the extinction time of the network is much more delicate. To this end, we performed, for small network sizes, exact simulations of the jump process in the case of the linear firing function $b(x)=\lambda \,x$. In the specific model we treat here, the particularly simple form of the dynamics of the variables $X_i(t)$ between spikes and the simplicity of the firing map b allows to derive the cumulative density function of the spikes:

$$\begin{aligned} \mathbb {P}(\tau _i \ge t) = \exp (-\lambda X_i (1-\exp (-t))) \end{aligned}$$

provided that $X_i(0)=X_i$. From this expression, one obtains the probability that neuron i stops firing $p_i=\exp (-\lambda X_i)$, and also the probability of firing at time t provided that the neuron does not stop firing. Therefore, although we deal with state-dependent Poisson processes, these formulae allow to simulate exactly the process and the extinction time, reached when all neurons stop firing.