Lectures in Theoretical Biophysics

its variables can still be interpreted in terms of physiological variables, since they have been derived from the physiological variables of the Hodgkin–Huxley model. The other way in which Fitzhugh derived these equations was by introducing one additional parameter in the original van der Pol oscillator equations which modifies them so that they have the qualitative features which Bonhoeffer had postulated, namely showing excitable and oscillatory behavior. The original van der Pol oscillator equations are symmetric in the sense that they are unaffected by a reversal of the sign of the variables. Since this symmetry is broken in the BvP model by the introduction of an additional parameter, the BvP model represents an important generalization of the van der Pol oscillator for the description of oscillatory phenomena. By studying the BvP system, we can thus learn something about physiological neuronal systems, and study a very important modification and generalization of the van der Pol oscillator dynamics. In the Bonhoeffer–van der Pol model, the dynamics of a single neuron is described by a system of two coupled differential equations ẋ1 = F1(x1, x2) = c · (x1 − x1/3 + x2 + z) ẋ2 = F2(x1, x2) = −1c · (x1 + bx2 − a). (8.1) According to Fitzhugh’s derivation of the BvP equations [?], x1 represents the negative transmembrane voltage and x2 is closely related to the potassium conductivity. The dynamical character of the solutions of this system of equations is determined by the parameter z, which represents the excitation of a neuron. In the absence of noise, z determines whether the system is an oscillator which periodically changes its voltage, or an excitable element which rests at a fixed voltage. The phase portrait for these two dynamical modes is shown in Figs. 8.1 and 8.2. For an analysis of the BvP system, we first want to determine the stationary point of the system. It can be obtained as the intersection point of the nullclines. The nullclines for the dynamical system April 23, 2000 Preliminary version