Abstract
A lumped model of neural activity in neocortex is studied to identify regions of multi-stability of both steady states and periodic solutions. Presence of both steady states and periodic solutions is considered to correspond with epileptogenesis. The model, which consists of two delay differential equations with two fixed time lags is mainly studied for its dependency on varying connection strength between populations. Equilibria are identified, and using linear stability analysis, all transitions are determined under which both trivial and non-trivial fixed points lose stability. Periodic solutions arising at some of these bifurcations are numerically studied with a two-parameter bifurcation analysis.
Highlights
Epilepsy is a neurological disease characterized by an increased risk of recurring seizures that affects about 1% of the world population
For fixed α1, we find that system can have one or two stable steady states
Multi-stability of two equilibria and two periodic solutions exists for α2 between H3 and PD3
Summary
Epilepsy is a neurological disease characterized by an increased risk of recurring seizures that affects about 1% of the world population. We notice the work by Shayer and Campbell [17] that studies a model very similar to the system (Equation 1) except for the fact that they choose the activation functions as odd functions They assumed that the connections between the nodes could be faster in one direction than in the other, and they studied the model’s dependency on this difference in time lags They are, to our knowledge, the only group that has performed a numerical bifurcation study of periodic orbits in two parameters for this type of model. We are primarily interested in the parameters related to connection strength as these may be amended with antiepileptic drugs These results will depend on the chosen values of the delays, we elaborate on their robustness under variations of these delays in the discussion. In the ‘Numerical bifurcation analysis’ section, we use software packages to determine (numerically) how the presence and stability of the bifurcating periodic solutions depend on the parameters α1 and α2
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