Network dynamics

In vivo cortical recording reveals that indirectly driven neural assemblies can produce reliable and temporally precise spiking patterns in response to stereotyped stimulation. This suggests that, despite being fundamentally noisy, the collective activity of neurons conveys information through temporal coding. Stochastic integrate-and-fire models offer a natural theoretical framework to study the interplay of intrinsic neural noise and spike timing precision. However, there are inherent difficulties to simulate their networks’ dynamics in silico with standard numerical discretization schemes. Indeed, evolving such networks requires temporally ordering every neuronal interaction, whereas the order of interactions is highly sensitive to the random variability of spiking times. To answer these issues, we designed an exact event-driven algorithm for the simulation of recurrent networks of perfect stochastic integrate-and- fire neurons with delayed Dirac-like interactions.

The two movies below illustrate the dynamics of two interacting neurons embedded in a larger network. The "synchronous picture" shows the instantaneous membrane voltage of each neuron fluctuating due to network interactions and internal noise. The "asynchronous picture" shows the progressive refinements of the next-spiking times of the two neurons.

In addition to being exact from the mathematical standpoint, our method is extremely efficient numerically. Our method is in fact a generalization of the well-known event-driven Gillespie algorithm employed to simulate the dynamics of chemical reactions. In the event-driven Gillespie algorithm, the concentration-dependent rates of each reaction are estimated to compute the time of the next reaction event and the subsequent update of the rates. Our algorithm leverages these ideas to a network of stochastic integrate-and-fire neurons: At each step, we simulate the next times at which each neuron would spike if they were independent and select the first-spiking event. This first-spiking event, in truth a real network event, is then used to update the next-spiking times of downstream neurons. As opposed to chemical reactions, the spiking times of integrate-and-fire neurons do not follow Poisson distributions, but are first-passage times of an underlying diffusion process describing the membrane voltage of neurons.

We utilize the above rigorous simulation framework to explore the dynamics of network of integrate-and-fire neurons for different synaptic connectivities, input drives, and learning dynamics. In the case of instantaneous interactions, the network dynamics can be analyzed as a diffusion process that is killed when hitting the a boundary of a high-dimensional quadrant, and reborn in the bulk of the quadrant according to rules prescribed by network interactions. We are interested in developing new effective theories to study the corresponding network dynamics in limits that depart from classical mean-field theory.