We study the stochastic dynamics of the Hindmarsh-Rose model of neuronal activity in the parametrical zone close to the Neimark-Sacker bifurcation. We show that in this zone, random disturbances transform the tonic spiking dynamic regime into the bursting one. This stochastic phenomenon is confirmed by the approximations of the probability density functions for the distribution of random trajectories as well as the interspike intervals statistics. For a quantitative analysis of the noise-induced bursting, we suggest and effectively apply a constructive approach based on the stochastic sensitivity function technique and the method of confidence domains.
Consider the stochastic variant of the model (
bz + "w_ ; here w is a standard Wiener process with E(w(t) w(s)) = 0; E(w(t) w(s))2 = jt sj, and the value " is a noise intensity.
Here, we focus on the parametric zone > 2 0:1603, where the unforced deterministic system (
Consider the value = 0:15. Here, the limit cycle is the attractor of the deterministic system (
Let us study the details of the noise-induced transition from tonic spiking regime to the bursting one by examining the changes of the distribution of random trajectories under increasing noise. Fig. 4 shows the changes of probability density distribution of z-coordinates of stochastic trajectories for = 0:15 in dependence on noise intensity. For small noise intensities, random states are localized near the deterministic limit cycle and have a sufficiently small dispersion. With an increase of noise, the dispersion of random states abruptly grows. This corresponds to the emergence of noise-induced torus oscillations.
A probability density distribution of random trajectories can be considered as a spacial characteristic for noise-induced torus bursting. As for an additional temporal (frequency) characteristic, interspike intervals (ISI) statistics can be used.
An estimation of interspike intervals distribution is a common method to study stochastic changes in oscillatory dynamics of neuron models. Such statistics of ISIs as the mean value m = h i, and the coefficient of p (
h variation (CV), CV = mm)2i , are widely applied.
Fig. 5 shows the overall mean value m (Fig. 5a) and the overall coefficient of variation CV (Fig. 5b) of ISIs for different parameter values under variation of the noise intensity. One can observe that for small noise, the mean ISI is almost constant and corresponds to the period of spiking limit cycle. The increase of the noise intensity causes the abrupt rise of the mean ISI due to the emergence of long ISIs corresponding to the quiescence phase in bursting regime. The plots of overall CV display the anti-coherence (the increase of variability of ISIs a) b) under random disturbances). This is also typical for the bursting dynamics. With the further increase of the noise intensity, the variability of ISIs decreases, which corresponds to the growth of the system coherence.
Thus, in the parametric zone > 2, where the limit cycle is the attractor of the deterministic system (
The emergence of the torus type stochastic oscillations is related to the peculiarities of the geometrical arrangement of deterministic trajectories near the limit cycle and its stochastic sensitivity.
Fig. 6 shows the deterministic trajectories started from different points in the vicinity of the limit cycle for = 0:15. The trajectories tend to the stable cycle, but the character of this movement can be different. One can determine two types of transient regimes in the phase space. In the first type, the trajectory tends to the limit cycle monotonically (see purple color in Fig. 6). In the second type of transient, the trajectory first goes far from the limit cycle, spend a long time in the vicinity of the unstable equilibrium, and then makes a long approach to the cycle (see blue color in Fig. 6). The type of the transient regime depends on the location of the initial point. Thus, there is some border surface between these transient regimes in the phase space. Let us define this border by term “pseudo-separatrix”.
The behavior of the system in the presence of random disturbances is also influenced by the stochastic sensitivity of attractors. To conduct the parametric analysis of the noise-induced torus bursting, we apply the stochastic sensitivity function (SSF) technique [20, 21].
Let us consider a general nonlinear system of stochastic differential equations:
dx = f (x) dt + " (x) dw(t):
(
Let the corresponding deterministic system (" = 0) have an exponentially stable limit cycle , defined by a T -periodic solution x(t) = x(t + T ).
Let t be a hyperplane that is orthogonal to the cycle at the point x(t) (0 t < T ). For this plane, in the neighborhood of the point x(t), a Gaussian approximation of the stationary probabilistic distribution of random states can be written [20] as: t(x; ") = K exp (x x(t))>W +(t)(x 2"2 x(t)) with the mean value mt = x(t) and the covariance matrix D(t; ") = "2W (t). Here, the matrix function W (t) is singular (det W = 0), and the sign "+"means a pseudoinversion. The matrix W (t) is a unique solution of the boundary problem
W_ = F (t)W + W F >(t) + P (t)S(t)P (t)
(
W (T ) = W (0);
W (t)r(t) = 0: F (t) =
(x(t)); S(t) = G(t)G>(t); G(t) = (x(t)); r(t) = f (x(t)); P (t) = Pr(t); Pr = I rr> r>r :
The stochastic sensitivity function W (t) characterises a sensitivity of a stable limit cycle to noise. In Fig. 7a, the nonzero eigenvalues 1;2(t) of the the matrix function W (t) for = 0:15 are plotted. One can observe that the stochastic sensitivity vary nonuniformly along the cycle.
To determine the stochastic sensitivity of the cycle as a whole, the stochastic sensitivity factor M = max 1(t) [0;T ] can be used. In Fig. 7b, the stochastic sensitivity factor M ( ) for limit cycles in the zone 2 ( 0:1603; 0:14) is plotted. One can observe that as the parameter approaches to the point = 0:1603 where the system undergoes the Neimark–Sacker bifurcation, the sensitivity of the limit cycle grows unlimitedly.
The SSF technique allows us to approximate the geometry of bundle of stochastic trajectories around the deterministic limit cycle. Eigenvalues and eigenvectors of the SSF matrix W (t) define a confidence ellipse located in the plane t with the center at the point x(t): (x x(t))>W +(t)(x x(t)) = 2q2"2; where the parameter q determines a fiducial probability P = 1 e q. With the given probability P , random states belong to the set specified by this confidence ellipse. A set of these ellipses for all points x(t) forms in the 3D-space a confidence torus around the limit cycle. Fig. 8a shows the confidence torus (a set of confidence ellipses) around the deterministic limit cycle for = 0:15, " = 0:0005 with fiducial probability P = 0:99.
Let us consider the mutual position of the confidence torus and pseudo-separatrix under different noise intensities. Consider the point of the limit cycle from the transition zone (i.e. the part of a limit cycle, from which the transition to the “bursting” zone of phase space occurs more frequently), and the a plane orthogonal to the limit cycle at this point. Let us construct the pseudo-separatrix line, which is an intersection of the pseudoseparatrix surface with the considered plane, and confidence ellipses in this plane. Fig. 8b displays a point of a) b) cycle from the transition zone, pseudo-separatrix for = 0:15, and the confidence ellipses for different noise intensities. For the sufficiently small noise intensity, a confidence ellipse is close to the deterministic cycle. With an increase of the noise intensity, the ellipse expands and intersects the pseudo-separatrix. Such intersection signalizes that with high probability, stochastic trajectories can go to the zone of the phase space where they form a stochastic torus. The noise intensity that corresponds to the intersection of the confidence ellipse with pseudo-separatrix can be used as an estimation of the critical value " . For = 0:15 we get the value " 0:003 which is in a good agreement with the results of the direct numerical simulations. 5
We studied the stochastic dynamics of the Hindmarsh–Rose model of neuronal activity in the parametrical zone close to the Neimark–Sacker bifurcation of the invariant torus generation. We showed that in this zone, random disturbances transform the tonic spiking dynamic regime into the bursting one. This stochastic phenomenon is confirmed by the approximations of the probability density functions for the distribution of random trajectories as well as the interspike intervals statistics. For a quantitative analysis of the noise-induced bursting, we suggested and applied a constructive approach based on the stochastic sensitivity function technique and the method of confidence domains. This method showed its reliability by a good agreement with the results of the direct numerical simulations.
The work was supported by Russian Science Foundation (N 16-11-10098).
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