Dynamical Systems in Neuroscience:

476 Synchronization (see www.izhikevich.com)Figure 10.22: Ioel Gil’evich Malkin (IoзlьGilьeviq Malkin, 1907-1958).where the T -periodic function Q is the solution to the linear “adjoint” equation˙Q = −{Df(x(t))} ⊤ Q , with Q(0) · f(x(0)) = 1 , (10.10)where Df(x(t)) ⊤ is the transposed Jacobian of f (matrix of partial derivatives) atthe point x(t) on the limit cycle, and the normalization condition can be replaced byQ(t) · f(x(t)) = 1 for any and hence all t (prove it). Here Q · f is the dot product oftwo vectors, which is the same as Q ⊤ f.Though this theorem looks less intuitive than the methods of Winfree and Kuramoto,it is actually more useful because (10.10) can be solved numerically quiteeasily. Applying the MATLAB procedure in Ex. 12 to the four oscillators in Fig. 10.3,we plot their functions Q in Fig. 10.23. It is not a coincidence that each componentof Q looks like PRC along the first or the second state variable, respectively, shown inFig. 10.6. Subtracting (10.9) from (10.8) or from (10.6), we conclude thatZ(ϑ) = grad ϑ(x) = Q(ϑ) ,(see also Ex. 7), so that we can determine the linear response function of the phasemodel using any of the three alternative methods: via PRCs, via isochrons, or solvingthe adjoint equation (10.10). This justifies why many refer to the function as just PRC,implicitly assuming that it is measured to the infinitesimal stimuli and then normalizedby the stimulus amplitude.10.2.4 Measuring PRCs experimentallyIn Fig. 10.24 we exploit the relationship (10.9) and measure the infinitesimal PRCsof a layer 5 pyramidal neuron of mouse visual cortex. First, we stimulate the neuronwith 40 pA dc-current to elicit periodic spiking. Initially, the firing period starts at50 ms, and then relaxes to the averaged value of 110 ms (Fig. 10.24a). The standardmethod of finding PRCs consists in stimulating the neuron by brief pulses of currentat different phases of the cycle and measuring the induced phase shift, which couldbe approximated by the difference between two successive periods of oscillation. Themethod works fine in models, see Ex. 5, but should be used with caution in real neuronsbecause their firing is too noisy, as we demonstrate in Fig. 10.24b. Thus, one needs