Mathematical modeling of electrophysiological activity in epilepsy

In computational neuroscience, the Wilson–Cowan model describes the dynamics of interactions between populations of very simple excitatory and inhibitory model neurons. It was developed by Hugh R. Wilson and Jack D. Cowan and extensions of the model have been widely used in modeling neuronal populations. The model is important historically because it uses phase plane methods and numerical solutions to describe the responses of neuronal populations to stimuli. Because the model neurons are simple, only elementary limit cycle behavior, i.e. neural oscillations, and stimulus-dependent evoked responses are predicted. The key findings include the existence of multiple stable states, and hysteresis, in the population response. In computational neuroscience, the Wilson–Cowan model describes the dynamics of interactions between populations of very simple excitatory and inhibitory model neurons. It was developed by Hugh R. Wilson and Jack D. Cowan and extensions of the model have been widely used in modeling neuronal populations. The model is important historically because it uses phase plane methods and numerical solutions to describe the responses of neuronal populations to stimuli. Because the model neurons are simple, only elementary limit cycle behavior, i.e. neural oscillations, and stimulus-dependent evoked responses are predicted. The key findings include the existence of multiple stable states, and hysteresis, in the population response. The Wilson–Cowan model considers a homogeneous population of interconnected neurons of excitatory and inhibitory subtypes. The fundamental quantity is the measure of the activity of an excitatory or inhibitory subtype within the population. More precisely, E ( t ) {displaystyle E(t)} and I ( t ) {displaystyle I(t)} are respectively the proportions of excitatory and inhibitory cells firing at time t. They depend on the proportion of sensitive cells (that are not refractory) and on the proportion of these cells receiving at least threshold excitation. Proportion of cells in refractory period (absolute refractory period r {displaystyle r} ) ∫ t − r t E ( t ′ ) d t ′ {displaystyle int _{t-r}^{t}E(t')dt'} Proportion of sensitive cells (complement of refractory cells) 1 − ∫ t − r t E ( t ′ ) d t ′ {displaystyle 1-int _{t-r}^{t}E(t')dt'} If θ {displaystyle heta } denotes a cell's threshold potential and D ( θ ) {displaystyle D( heta )} is the distribution of thresholds in the tissue, then the expected proportion of neurons receiving an excitation at or above threshold level per unit time is S ( N ¯ ) = ∫ 0 N ¯ ( t ) D ( θ ) d θ {displaystyle S({ar {N}})=int _{0}^{{ar {N}}(t)}D( heta )d heta } , where N ¯ ( t ) {displaystyle {ar {N}}(t)} is the mean integrated excitation at time t. The term 'integrated' in this case means that each neuron sums up (i.e. integrates) all incoming excitations in a linear fashion to receive its total excitation. If this integrated excitation is at or above the neuron's excitation threshold, it will in turn create an action potential. Note that the above equation relies heavily on the homogeneous distribution of neurons, as does the Wilson-Cowan model in general.