Izhikevich model

These dynamical equations can be injected in the streaming flux term of the conservation equation as they describe the evolution of a single neuron only driven by its own dynamics. So the complete form of the conservation equation describing the dynamics of a neural population and using the Izhikevich model to describe the evolution of a single element is ... 

Fig. 13.6 Some examples of the richness of the Izhikevich model (electronic version of the figure and reproduction permissions are freely available atwww.izhikevich.com).

In order to validate the reduced model (uncoupled population of Izhikevich neurons) we chose to perform comparisons with a direct simulation model. In this last model the internal state of each neuron is computed at each time step (with a forth-order Runge-Kutta method) using the equations of the Izhikevich model so we have complete access to individual information as opposed to the population density formalism where only the states distribution can be computed. The simulation parameters were 50 000 Izhikevich neurons in tonic spiking mode (a = 0.02, b = 0.2, c = —65, d = 6 as provided in [29]), a gaussian form of p(v, u, t) at t =0 and a constant input current of / =60 qA was applied to all neurons. The firing rate and the mean membrane potential (M M P) were computed at each time for the two methods during 15 ms. 

This model should allow studying the validity of several mechanisms of DBS that have been proposed over the past few years, like the excitation or blockade of (1) neurons in the stimulation area, (2) neurons projecting to the stimulated area via fibers of passage. Another hypothesis is the differential effects of stimulation between the soma and axon of neurons [41] and this could be tested by modifying the Izhikevich model. Frequency dependence, one of the most mysterious aspects of DBS, could be explored too as well as the possibility of modulation of synaptic kinetics by DBS. 

Modolo J., Garenne A., Henry J., Beuter A. A new population density approach using the Izhikevich model (in preparation for the / Comput Neurosci). 

Comparison of the integration of this last conservation equation with the direct simulation of an uncoupled neural population with a given distribution of states (v, u) at the initial time provides a validation for the simplified version of the model. Then adding the imposed flux accounting for connectivity allows us to simulate a large population of Izhikevich neurons with a given pattern of connectivity (number of afferents per neuron and delays kernel). 

Izhikevich E. M. Simple model of spiking neurons. IEEE Trans Neural Network, 2003, 14,1569-1572. 

Izhikevich E. M. Which model to use for cortical spiking neurons IEEE Trans Neural Netw, 2004, 25,1063-70. 

F. Hoppenstead and E.M. Izhikevich, Canonical neural models. Brain Theory and Neural Networks, 2 Ed., MIT Press, Cambridge, MA, (2002). 

---