Quentin Cormier

Numerical Simulations

Two interactive applications to explore mean-field models of spiking neurons. In the equations below, \(V_t\) is the membrane potential and \(N_t\) counts spikes, with intensity \(f(V_{t-})\). Each spike resets the potential to zero; \(J\) controls the strength of the interaction.

2D spiking neurons with synaptic depression

This is a numerical illustration of this work. One models a coupling between the membrane potential and a synaptic fatigue mechanism \(X_t \in [0, 1]\). The synapse recover between spikes and decrease by a fraction \(U\) at each spike, reducing the neuron's contribution to the network. The McKean-Vlasov model is

\[ \begin{aligned} \mathrm{d}V_t &= [b(V_t)+J a_t]\,\mathrm{d}t - V_{t-}\,\mathrm{d}N_t, \\ \mathrm{d}X_t &= \frac{1-X_t}{\tau}\,\mathrm{d}t - U X_{t-}\,\mathrm{d}N_t, \\ a_t &= \mathbb{E}[X_t f(V_t)], \qquad b(v)=\lambda(m-v). \end{aligned} \]

Here \(N_t\) is a point processes of stochastic intensity \(f(V_{t-})\), \(\tau\) is the recovery time and \(f\) is a firing-rate function (for example, \(f(v)=v^2\)). Simulate spike trains and population activity, compute invariant measures, and explore their stability.

Open the 2D spiking neuron app →

3D bifurcation explorer

Explore the Hopf bifurcations in the parameter space \((J,m,\beta)\) of following stochastic Integrate-and-fire model:

\[ \begin{aligned} \mathrm{d}V_t &= (m-V_t+J r_t)\,\mathrm{d}t - V_{t-}\,\mathrm{d}N_t, \\ r_t &= \mathbb{E}[f(V_t)], \\ f(v) &= \frac{1}{\beta}\,\mathbf{1}_{\{v\geq 1\}}, \qquad \beta>0. \end{aligned} \]

Neurons fire at rate \(1/\beta\) above threshold. Select a parameter point and run a particle simulation to inspect the firing rate, mean potential and spike trains. On the Hopf surface, the color indicates the oscillation frequency \(y\), with corresponding period \(T=2\pi/y\).

Open the 3D bifurcation explorer →