Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 4771Andronov-Hopf oscillator1Q 1 ( )Q( )Q 2 ( )Q 2Q 1 phase,1Q 2Q 1van der Pol oscillatorQQ( )1 ( )1Q 2 ( )0000Q 1 Q 1-1-1-1-1 0 1 0 2 4 6-1-2 0 2 0 2phase,4 6Q 2 Q 21000.2 Q 1 ( )800Q( ) 0.01Q 2 ( )Q( )I Na +I K -model (Class 1) I Na +I K -model (Class 2)-1006005-20400-30-0.22000-40-0.1 0 0.1 0.2-0.40.01Q 2 ( )0 2 4 6phase,0-200-5 0 5-5Q 1 ( )0 1 2 3phase,Figure 10.23: Solutions Q = (Q 1 , Q 2 ) to adjoint problem (10.10) for oscillators inFig. 10.3.to apply hundreds if not thousands of pulses and then average the resulting phasedeviations (Reyes and Fetz 1993).Starting with time 10s we inject a relatively weak noisy current εp(t) that continuouslyperturbs the membrane potential (Fig. 10.24c) and hence the phase of oscillation(the choice of p(t) is important; its Fourier spectrum must span a range of frequenciesthat depends on the frequency of firing of the neuron). Knowing εp(t), the momentsof firing of the neuron, i.e., zero crossings ϑ(t) = 0, and the relationship˙ϑ = 1 + PRC (ϑ)εp(t) ,we solve the inverse problem for the infinitesimal PRC (ϑ) and plot the solution inFig. 10.24d. As one expects, the PRC is mostly positive, maximal just before the spikeand almost zero during the spike. It would resemble the PRC in Fig. 10.23 (Q 1 (ϑ) inClass 1) if not for the dip in the middle, for which we have no explanation (probably itis due to overfitting). The advantage of this method is that it is more immune to noise,because intrinsic fluctuations are spread over the entire p(t) and not concentrated atthe moments of pulses, unless of course p(t) consists of random pulses, in which casethis method is equivalent to the standard one. The drawback is that we need to solvethe equation above, which we do in Ex. 13 using an optimization technique.