Dynamical Systems in Neuroscience:

246 Excitabilityspikeexcitatory pulsesspikeinhibitory pulsesspikesmallPSPHodgkin-Huxley modeln+m+hspikeVresonantnon-resonant21211 22121Figure 7.22: (Left) Projection of trajectories of the Hodgkin-Huxley model on a plane.(Right) Phase portrait and typical trajectories during resonant and non-resonant responseof the model to excitatory and inhibitory doublets of spikes. Modified fromIzhikevich (2000).out in this case. Similarly, the spikes cancel each other when the interpulse period is15 ms, which is 60 % greater than the natural period. The same phenomenon occursfor inhibitory synapses, as we illustrate in Fig. 7.21. Here the second pulse increases(decreases) the amplitude of oscillation if it arrives during the falling (rising) phase.We study the mechanism of such frequency preference in Ex. 4, and present itsgeometrical illustration in Fig. 7.22. There, we depict a projection of the phase portraitof the Hodgkin-Huxley model having a stable focus equilibrium. The model does nothave a true threshold, as we discuss in Sect. 7.2.4. To fire a spike, a perturbationmust push the state of the model beyond the shaded figure that is bounded by twotrajectories, one of which corresponds to a small post-synaptic potential (PSP), whilethe other corresponds to a spike.Fig. 7.22, right. depicts responses of the model to pairs of pulses, called doublets.Pulse 1 in the excitatory doublet shifts the membrane potential from the equilibriumto the right, thereby initiating a subthreshold oscillation. The effect of pulse 2 dependson its timing: If it arrives when the trajectory finishes one full rotation around theequilibrium, then it pushes the voltage variable even more to te right, beyond theshaded area into the spiking zone, and the neuron fires an action potential. In contrast,if it arrives too soon, the trajectory does not finish the rotation, and it is still to theleft of the equilibrium. In this case, pulse 2 pushes the state of the model closer tothe equilibrium, thereby canceling the effect of pulse 1. Similarly, the effect of aninhibitory doublet depends on the interspike period between the inhibitory pulses. Ifthe interpulse period is near the natural period of damped oscillations, pulse 2 arriveswhen the trajectory finishes one full rotation, and it adds to pulse 1, thereby firing theneuron. If it arrives too soon or too late, it cancels the effect of pulse 1.Quite often, the frequency of subthreshold oscillations depends on their amplitudes,e.g., oscillations in the Hodgkin-Huxley model slow down as they become larger. In this