Spectral resonance in Fitzhugh-Nagumo neuron system: relation with stochastic resonance and its role in EMG signal characterization

Abstract

This paper examines the existence of spectral resonance in the Fitzhugh-Nagumo (FHN) system driven by periodical signal and unbounded noise having Gaussian distribution. It is newly revealed that if the inter-spike-interval (ISI) distribution is accumulated on a single cluster, there exists a dual relationship between stochastic resonance and spectral resonance determined by commonly used metric normalized standard deviation of ISI. Furthermore, the ISI distribution is also concentrated on more than one cluster depending on different driving signal frequency. Consequently, the apparent regular spiking behavior is observed to occur at specified driving signal frequencies which result in a local minimum in entropy function indicating spectral resonance. Therefore it is proposed that occurrence of spectral resonance strongly depends on the shape of ISI distribution tuned by the stochastic and deterministic driving signal parameters and conventional metrics may not indicate entire resonance behavior. Correspondingly, the entropy function is utilized in this paper as an alternative metric to enable the detection of the spectral resonance occurrence. The ISI distribution obtained from the FHN system is investigated to relate the real electromyography (EMG) measurements under different conditions such as myokymia and neuromyotonia. It is seen that ISI distribution observed from myokymic EMG exhibits notably close behavior as in the case of spectral resonance generated by FHN whereas a wider distribution is monitored in the case of neuromyotonia. It is contributed that the modeling and parameterization based on ISI distribution can be potentially used to identify different neural activities.

Keywords: Fitzhugh–Nagumo neuron system; Myokymia; Spectral resonance; Stochastic resonance.

Fig. 1

Determined subthreshold and suprathreshold region in FHN model with respect to $A$ and $f_i$ pairs. The parameters can be chosen in subthreshold region according to the determined boundary region in order to achieve spectral resonance apparently in presence of noise injection with minimum but sufficient intensity

Fig. 2

Determination of optimal time delay $w_{\mathit{opt}}$ from $\psi \left[n\right]$ for TKEO based spike detection. The time delay corresponding first local maximum yields an information about optimum value to determine TKEO result more accurately

Fig. 3

a Membrane potential $x\left[n\right]$ from the FHN system, b TKEO function $\varphi \left[n\right]$ together with threshold $T$(dashed red line). (Color figure online)

Fig. 4

The peak point determined by TKEO process after determining optimum time delay $w_{\mathit{opt}}$

Fig. 5

a Variation of entropy $H$ in terms of $f_i$; the local minimums are located at $f_1=0.2154\mathrm{Hz},$ $f_2=0.4375\mathrm{Hz},$ and $f_3=0.7017\mathrm{Hz},$ b Variation of normalized deviation $C$ in terms of $f_i$, $f_4=0.1194\mathrm{Hz}$

Fig. 6

The membrane potential and the corresponding ISI distribution with respect to different input frequencies; a and b for $f_1=0.2154\mathrm{Hz},$ c and d for $f_2=0.4375\mathrm{Hz},$ e and f for$f_3=0.7017$, $\left(D,=,0.25\right)$. As the frequency of periodic exciting signal increases, ISI distribution becomes to widen and include multi-clustered structure

Fig. 7

a The membrane potential $x\left(t\right)$ ($f_4=0.1194\mathrm{Hz}$, $D=0.25$), b ISI distribution concentrated on single cluster

Fig. 8

Illustration of stochastic resonance behavior in terms of $C$ as a function of noise intensity $D$. When the resonance location is investigated, it is seen to be one-to-one correspondence between noise intensities and specified frequencies shown in Fig. 5b

Fig. 9

a The EMG signal in case of myokymia, b ISI distribution in myokymic EMG c the EMG signal in case of neuromyotonia d ISI distribution in neuromyotonic EMG