Mode-locking behavior of Izhikevich neurons under periodic external forcing

Many neurons in the auditory system of the brain must encode periodic signals. These neurons under periodic stimulation display rich dynamical states including mode locking and chaotic responses. Periodic stimuli such as sinusoidal waves and amplitude modulated sounds can lead to various forms of $n:m$ mode-locked states, in which a neuron fires $n$ action potentials per $m$ cycles of the stimulus. Here, we study mode-locking in the Izhikevich neurons, a reduced model of the Hodgkin–Huxley neurons. The Izhikevich model is much simpler in terms of the dimension of the coupled nonlinear differential equations compared with other existing models, but excellent for generating the complex spiking patterns observed in real neurons. We obtained the regions of existence of the various mode-locked states on the frequency-amplitude plane, called Arnold tongues, for the Izhikevich neurons. Arnold tongue analysis provides useful insight into the organization of mode-locking behavior of neurons under periodic forcing. We find these tongues for both class-1 and class-2 excitable neurons in both deterministic and noisy regimes.

Figure 1

F-I curves for (a) class-1 and (b) class-2 neurons with parameters given in the text.

Figure 2

Arnold tongue diagram for a class-1 Izhikevich neuron with the F-I curve shown in Fig. 1, driven by an external sinusoidal forcing. The DC current is $62\mu \mathrm{A}/{\mathrm{cm}}^2$.

Figure 3

First example: $3:2$ mode-locked pattern.

(a) Time series diagram of a sinusoidal stimulus with amplitude of $A=45\mu \mathrm{A}/{\mathrm{cm}}^2$ and frequency of $f=7.5$ Hz (blue) and the corresponding spike pattern (red). (b) Frequency spectrum of the spike pattern. (c) Phase-space diagram.

Figure 4

Second example: $2:1$ mode-locked pattern. (a) Time series diagram of a sinusoidal stimulus with amplitude of $A=20\mu \mathrm{A}/{\mathrm{cm}}^2$ and frequency of $f=5$ Hz (blue) and the corresponding spike pattern (red). (b) Frequency spectrum of the spike pattern. (c) Phase-space diagram.

Figure 5

Arnold tongues diagram for a class-2 Izhikevich neuron driven by an external sinusoidal forcing that corresponds to the neuron with the F-I curve and parameters shown in Fig. 1. The DC current is $120\mu \mathrm{A}/{\mathrm{cm}}^2$.

Figure 6

First example: $3:2$ mode-locked pattern.

(a) Time series diagram of a sinusoidal stimulus with amplitude of $A=110\mu \mathrm{A}/{\mathrm{cm}}^2$ and frequency of $f=75$ Hz (blue) and the corresponding spike pattern (red). (b) Frequency spectrum of the spike pattern. (c) Phase-space diagram.

Figure 7

Second example: $2:3$ mode-locked pattern. (a) Time series diagram of a sinusoidal stimulus with amplitude of $A=120\mu \mathrm{A}/{\mathrm{cm}}^2$ and frequency of $f=180$ Hz (blue) and the corresponding spike pattern (red). (b) Frequency spectrum of the spike pattern. (c) Phase-space diagram.

Figure 8

(a) Time series and (b) phase-space diagram of the system corresponding to $A=110\mu \mathrm{A}/{\mathrm{cm}}^2$ and $f=36$ Hz in the class-2 neuron. Referring to Fig. 5 shows this selection of stimulus parameters leads to $3:1$ mode locking. This can be seen also in the solution and formation of limit cycle here.

Figure 9

(a) Time series and (b) phase-space diagram of the system corresponding to $A=110\mu \mathrm{A}/{\mathrm{cm}}^2$ and $f=35$ Hz in the class-2 neuron. The solution is not stable any more, which leads to disappearance of mode-locking and periodic decay in phase space.

Figure 10

(a) Spike trains of a $1:1$ mode-locked state without noise and (b) the corresponding ISIs. (c) After adding the white Gaussian noise with $\mu =0$ and ${\sigma }^2=5$, the mode-locking pattern becomes less stable and (d) the corresponding ISIs no longer overlap on a well-defined point.

Figure 11

(a) Spike trains of a $2:1$ mode-locked state without noise and (b) the corresponding ISIs. (c) After adding the white Gaussian noise with $\mu =0$ and ${\sigma }^2=5$, the mode-locking pattern becomes less stable and (d) the corresponding ISIs no longer overlap on two well-defined points.

Figure 12

The unit circle that is used to find the smallest phase $\varphi$. The window $\delta \varphi$ is defined by the user and indicates the radius of the cluster under study in the ISIs; Fig. 11.

Figure 13

VS-based Arnold tongue diagram of the class-1 neuron in the presence of noise with different strengths (a) $\mu =0$, ${\sigma }^2=2$ and (b) $\mu =0$, ${\sigma }^2=5$. Time step for computing the Izhikevich model is 0.05 ms. The color code represents the amount of vector strength. Note how the different mode-locked states lose their stabilities with the addition of noise compared with Fig. 2.

Figure 14

VS-based Arnold tongue diagram of the class-2 neuron in the presence of noise with different strengths (a) $\mu =0$, ${\sigma }^2=2$ and (b) $\mu =0$, ${\sigma }^2=5$. Time step for computing the Izhikevich model is 0.05 ms. The color code represents the amount of vector strength. Note how the different mode-locked states lose their stabilities with the addition of noise compared with Fig. 5.

Figure 15

(a) Arnold tongues for the noisy class-2 neuron in a broader range of amplitude and frequency of periodic stimulus. (b) Zoomed-in region on the left side of panel (a) shown with a red rectangle. The tongues are entangled at high amplitudes. (c) Zoomed-in part of $1:2$ tongue shown with a red rectangle in panel (a).