Spike train statistics for consonant and dissonant musical accords in a simple auditory sensory model

The phenomena of dissonance and consonance in a simple auditory sensory model composed of three neurons are considered. Two of them, here so-called sensory neurons, are driven by noise and subthreshold periodic signals with different ratio of frequencies, and its outputs plus noise are applied synaptically to a third neuron, so-called interneuron. We present a theoretical analysis with a probabilistic approach to investigate the interspike intervals statistics of the spike train generated by the interneuron. We find that tones with frequency ratios that are considered consonant by musicians produce at the third neuron inter-firing intervals statistics densities that are very distinctive from densities obtained using tones with ratios that are known to be dissonant. In other words, at the output of the interneuron, inharmonious signals give rise to blurry spike trains, while the harmonious signals produce more regular, less noisy, spike trains. Theoretical results are compared with numerical simulations.

Figure 1

The investigated model. Sensory neurons $N_1$ and $N_2$ are driven by subthreshold sinusoidal signals with different frequencies. Spike trains of sensors are received by the interneuron $N_3$. All three neurons are perturbed by independent noise sources ${\xi }_1\left(t\right)$, ${\xi }_2\left(t\right)$ and ${\xi }_3\left(t\right)$.

Figure 2

(a) Typical behavior of the membrane potential $v_i\left(t\right)$ of sensory neurons versus time for a noise realization. (b) ISI distribution of the sensory neurons $\left(\Omega =0.6\right)$. The highest probability of a spike after $t=0$ is near one period of external force $\left(t=10.47\right)$. The probability of firing after two, three, etc. periods decreases exponentially. (c) Typical behavior of the membrane potential $v_3\left(t\right)$ of the interneuron versus time for the same noise realization. Here are well visible the refractory state (ref), characterized by the refractory time $T_{ref}$, and the noisy background (bg) during the relaxation time $T_{relax}$. The values of the system parameters are ${\mu }_1={\mu }_2=1$, ${\mu }_3=0.3665$ (relaxation parameters), $D_1=D_2=D_3=1.6\times {10}^{-3}$ (noise intensities), $k_1=k_2=0.98$ (coupling coefficients), $v_1^0=v_2^0=0$, $v_3^0=-1$, (reset values of the membrane potentials), and $v_{th}=1$ (threshold value of the membrane potential); $A_1=1.084$, $A_2=1.115$, ${\Omega }_1=0.4$, and ${\Omega }_2=0.42$ (amplitudes and frequencies of the periodical driving forces); and $T_{max}={10}^5$ (maximum output time length of the time series generated from the system). From Eqs. (2, 14), we have $T_{ref}=6.28$ and $T_{relax}=8.73$.

Figure 3

Probabilities ${\Phi }_i\left(k_i,k_j,t-t^{\prime }\right)$ as functions of the time difference $\left(t-t^{\prime }\right)$ for weak and strong connectivity. (a) Weak connectivity. The values of the coupling coefficients are: $k_1=0.6$, $k_2=0.7$. The probabilities relax quite rapidly to the zero value. Only two incoming spikes very close in time should be able to fire the interneuron. (b) Strong connectivity. The values of the coupling coefficients are: $k_1=0.94$, $k_2=0.98$. The time dependent probabilities relax slowly toward the values of conditional probabilities of “separate” spikes. Two consecutive incoming spikes from sensors have more time and greater probability to fire the interneuron. The values of the other parameters are: $D=1.6\times {10}^{-3}$, $\mu =0.3665$, and $v_{th}=1$.

Figure 4

FPTPDs of three neurons in states 0 and 1. This is the example of the probable transition between two states for the case of input sinusoids with frequencies related by ratio 4/3 (the perfect fourth accord). ${\rho }_2\left(t\right)$ is shifted in state 1 in comparison with ${\rho }_2\left(t\right)$ in state 0. As a result, ${\widehat{\rho }}_3^{\left(0\right)}\left(t\right)$ and ${\widehat{\rho }}_3^{\left(0\right)}\left(t\right)$ are different. All possible states of the interneuron are: state 0, state 1, …, and state 5. The peaks of ${\rho }_1\left(t\right)$ and ${\rho }_2\left(t\right)$ are marked by numbers in circles in order to establish the correspondence between them and the interneuron states.

Figure 5

Interspike interval distributions of the consonant accords: octave (2/1), perfect fifth (3/2), major third (5/4), and minor third (6/5). Under each picture there is the ratio of frequencies $\left(m/n\right)$ and the name of the accord used in the common musical terminology. All curves are obtained through the direct numerical simulation of the Eq. (1) with ${\Omega }_2=0.6$, $A_2=1.165$, ${\Omega }_1=\left(m/n\right){\Omega }_2$, and $A_1$ chosen according to the subthreshold input sinusoidal signal condition of Eq. (6). Specifically $A_1=1.52$ for octave (2/1), $A_1=1.325$ for perfect fifth (3/2), $A_1=1.243$ for major third (5/4), and $A_1=1.222$ for minor third (6/5). The other values of parameters are ${\mu }_1={\mu }_2=1$, ${\mu }_3=0.3665$, $D_1=D_2=D_3=1.6\times {10}^{-3}$, $k_1=k_2=0.98$, $v_1^0=v_2^0=0$, $v_3^0=-1$, and $v_{th}=1$.

Figure 6

Interspike interval distributions of the dissonant accords: major second (9/8), minor seventh (16/9), minor second (16/15), and augmented fourth (45/32). Under each picture there is the ratio of frequencies $\left(m/n\right)$ and the name of the accord used in the common musical terminology. All curves are obtained through the direct numerical simulation of the Eq. (1) with ${\Omega }_2=0.6$, $A_2=1.165$, ${\Omega }_1=\left(m/n\right){\Omega }_2$, and $A_1$ chosen according to the subthreshold input sinusoidal signal condition Eq. (6). Specifically $A_1=1.2$ for major second (9/8), $A_1=1.436$ for minor seventh (16/9), $A_1=1.17$ for minor second (16/15), $A_1=1.305$ for augmented fourth (45/32). The other values of parameters are the same as in Fig. 5.

Figure 7

ISI distributions of the output neuron for different accords. Solid lines are the theoretical results obtained from Eqs. (22, 25, 27). Dashed lines are the distribution obtained throughout the direct numerical simulation of system (1). The parameters are ${\mu }_1={\mu }_2=1$, $\mu =0.3665$, $k_1=k_2=0.97$, and $D_1=D_2=D=1.6\times {10}^{-3}$. Perfect fourth: $A_1=1.165$, ${\Omega }_1=0.6$, $A_2=1.085$, and ${\Omega }_2=0.45$. Minor third: $A_1=1.125$, ${\Omega }_1=0.54$, $A_2=1.085$, and ${\Omega }_2=0.45$. Major second: $A_1=1.2$, ${\Omega }_1=0.675$, $A_2=1.165$, and ${\Omega }_2=0.6$.

Figure 8

Illustration for the procedure of minimal peak distance obtaining.