Handbook Of Brain Activity by Ivana Suchy

By Ivana Suchy

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This form of non-identical synchronization is known as g eneralized chaotic synchronization (Afraimovich et al. 1986, Rulkov et al. 1995). Further details and examples of more complex behaviors – such as intermittency, scale-free dynamics and travelling waves - can be found in Breakspear et al. (2003, 2005). This concludes our survey of basic, small-scale neural systems. We hope to have illustrated the power of combining analysis and geometry in elucidating some of the fundamental properties of neurons.

However, rather than further reducing the Hodgkin-Huxley model, the Hindmarsh-Rose (1984) model introduces an extra property. The system is given by, dx = y − ax3 + by 2 − z + I, dt dy = c − dx2 − y, dt dz = r [s (x − x0 ) − z] , dt (41) (42) (43) When r = 0, the third variable plays no role and the system reduces to a variation of a FitzHugh Nagumo model – that is, a two dimensional spiking neuron with a simple algebraic form: An example is given in Fig. 22. However, setting r > 0 but small has the effect of introducing the third variable into the dynamics.

Arrows show representative vector field. (b) Long (magneta) and short (yellow) heteroclines of stable node and saddle In Fig. 16 is shown representative orbits of this system. Three “subthreshold” (green) and three “suprathreshold” (red) orbits are shown. In the latter case, the neuron depolarizes before returning to its resting state. It should be noted that this threshold depends not only on the initial membrane potential V but also the initial K+ membrane conductance. The separatrix between sub- and supra-threshold is constituted by the inset of the saddle point (not shown).

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