# Exact neural fields incorporating gap junctions

### Patterns and Simulations

By Carlo R. Laing
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We consider networks of quadratic integrate-and-fire neurons coupled via both chemical synapses and gap junctions. After transforming to “theta neuron” coordinates, a network’s governing equations are of a form amenable to the use of the Ott/Antonsen ansatz. This ansatz allows us to derive an exact description of a network’s dynamics in the limit of an infinite number of neurons. For an all-to-all connected network we derive a single (complex) ordinary differential equation while for spatially extended networks we derive neural field equations (nonlocal partial differential equations). We perform extensive numerical analysis of the resulting equations, showing how the presence of gap junctional coupling can destroy certain spatiotemporal patterns such as stationary “bump” solutions and create others such as travelling waves and spatiotemporal chaos. Our results provide significant insight into the effects of gap junctions on the dynamics of networks of Type I neurons.
 Author Institutional Affiliation Massey University, Auckland, New Zealand Author Email [email protected] Author Postal Mail Institute of Natural and Mathematical Sciences, Massey University, Private Bag 102-904 NSMC, Auckland, New Zealand Notes Published version: http://epubs.siam.org/doi/abs/10.1137/15M1011287 Keywords neural field, quadratic integrate-and-fire, gap junction, Ott/Antonsen, bifurcation, theta neuron

• Bifurcations induced by gap junctions(.png, 24.82 KB) - 3193 download(s) Bifurcations in an all-to-all coupled network of theta neurons. $$g$$ = gap junction coupling strength, $$\kappa$$ = synaptic coupling strength. Solid black curves: saddle-node bifurcations of fixed points; dashed red curve: homoclinic bifurcation; dash-dotted blue curve: Hopf bifurcation. There are stable periodic orbits in regions C and E.
• Hopf bifurcation of a "bump"(.png, 482.33 KB) - 3182 download(s) Hopf bifurcation of a bump state. Instantaneous frequency is shown colour-coded (the maximum is trucated). Gap junction coupling strength $$g$$ is switched from $$0$$ to $$0.6$$ at $$t=20$$.
• Long-time average firing frequency of bump(.png, 189.29 KB) - 3182 download(s) Long-time average firing frequency of the bump solution in previous figure. Left: $$g=0$$, right: $$g=0.6$$.
• Spatiotemporal chaos(.png, 1004.18 KB) - 3189 download(s) Spatiotemporal chaos caused by the inclusion of gap junctions. Top: $$| z|$$ and middle: $$\sin(\arg(z))$$, where $$z$$ is the Kuramoto order parameter. Bottom: simulation of an equivalent discrete system of 4096 neurons. ($$\sin{\theta_i}$$ is shown.)
• Hopf bifurcation of a front(.png, 306.32 KB) - 3189 download(s) Hopf bifurcation of a front in a network with purely positive coupling, caused by gap junction coupling. Top: Re($$z$$). Bottom: Im($$z$$), where $$z$$ is usual Kuramoto order parameter. $$g$$ (strength of gap junction coupling) is switched from $$0$$ to $$0.12$$ at $$t=100$$. Neurons are gap junction coupled to those within a distance of $$2.5$$ spatial units either side.