14 The Integral-equation Approach

14.7 Summary

Relating the microscopic level of single neurons to the macroscopic level of neuronal population, the integral equation approach offers an interpretation of neuronal activity in terms of the inter-spike interval distribution. The integral approach can be related to partial differential equations. The formulation of partial differential equations with refractory densities exhibits a close formal relation with the membrane potential density equations of Ch. 13. In a direct comparison of the two theories, the first one developed in Ch. 13, the second here, it turns out that the integral equation approach is particularly useful to model populations of neurons that have multiple intrinsic time scales in refractoriness, synapses or adaptation and escape noise as the model of stochasticity.

Population equations can be formulated for several coupled populations. At the steady state, the population can be in a state of asynchronous and irregular firing, but the stability of these solutions against emergence of oscillations needs to checked. Stability can be analyzed using the linearization of the integral equations around a stationary state.

Heterogeneity in the population can be treated as slow noise in the parameters and finite size effects can be analyzed and included in the numerical integration scheme.