^{1,a)}and Jordi García-Ojalvo

^{1,b)}

### Abstract

We present a physiologically plausible binaural mechanism for the perception of the pitch of complex sounds via ghost stochastic resonance. In this scheme, two neurons are driven by noise and a different periodic signal each (with frequencies and , where ), and their outputs (plus noise) are applied synaptically to a third neuron. Our numerical results, using the Morris–Lecar neuron model with chemical synapses explicitly considered, show that intermediate noise levels enhance the response of the third neuron at frequencies close to , as in the cases previously described of ghost resonance. For the case of an inharmonic combination of inputs ( and ) noise is also seen to enhance the rates of most probable spiking for the third neuron at a frequency . In addition, we show that similar resonances can be observed as a function of the synaptic time constant. The suggested ghost-resonance-based stochastic mechanism can thus arise either at the peripheral level or at a higher level of neural processing in the perception of pitch.

The perception and processing of environmental complex signals resulting from the combination of multiple inputs is a nontrivial task for the nervous system. In many species, solving efficiently this sensory problem could have an evolutionary payoff. A classical example is the perception of the pitch of complex sounds by the auditory system, the mechanism of which remains controversial. Recently, a mechanism for the perception of pitch has been proposed on the basis of the so-called ghost stochastic resonance. Under this paradigm, an appropriate level of noise yields an optimal subharmonic neural response to a combination of two or more harmonic signals that lack the fundamental frequency, which is nevertheless perceived by the system. The original proposal concentrated in the peripheral level of the perception process, by considering the case of a simple monoaural presentation of the complex signal. On the other hand, it is known that complex sounds are also perceived when its two constituent tones are presented binaurally (i.e., one in each ear). Thus, the question that remains is whether ghost stochastic resonance can participate in detecting this “virtual” dichotic pitch at a higher level of processing. In this paper we present, on the basis of numerical simulations, a plausible mechanism for the binaural perception of the pitch of complex signals via ghost stochastic resonance. In this scenario, each of the two input tones drives a different noisy neuron (corresponding to detection in the left–right auditory pathways), and together they drive a third noisy neuron that perceives the missing fundamental. In this way, the same basic mechanism of ghost reso-nance can explain pitch perception occurring at both the peripheral and a higher processing level.

We thank Dante R. Chialvo for guidance and useful comments on the paper. We also thank Claudio Mirasso for helpful remarks, and two anonymous referees whose constructive criticisms have led to qualitative improvements in the manuscript. We acknowledge financial support from MCyT-FEDER (Spain, Project Nos. BFM2002-04369 and BFM2003-07850), and from the Generalitat de Catalunya. P.B. acknowledges financial support from the Fundación Antorchas (Argentina).

I. INTRODUCTION

A. Pitch perception by single neurons

B. Signal integration and processing of distributed inputs

II. MODEL DESCRIPTION

A. Neuron model

B. Synapses model

III. THE CASE OF DISTRIBUTED HARMONIC COMPLEX SIGNALS

A. Deterministic case

B. Stochastic case

C. Quantifying the coherence detection efficiency

D. Role of synaptic coupling

IV. THE INHARMONIC CASE

V. CONCLUSIONS

### Key Topics

- Nerve cells
- 38.0
- Sound discrimination
- 17.0
- Binaural hearing
- 13.0
- Stochastic processes
- 13.0
- Pitch
- 11.0

## Figures

Deterministic response to a distributed harmonic complex signal. The membrane potential for the three neurons is shown: (a) and (b) Input neurons, (d) processing neuron. The synaptic current acting on neuron 3, , is shown in plot (c). The two input neurons are fed with two sinusoidal signals of amplitudes , , and periods , , respectively (which gives a ghost resonance of ). The bias current for all three neurons is , the synaptic coupling between input and processing neurons is and . All noise amplitudes are zero, .

Deterministic response to a distributed harmonic complex signal. The membrane potential for the three neurons is shown: (a) and (b) Input neurons, (d) processing neuron. The synaptic current acting on neuron 3, , is shown in plot (c). The two input neurons are fed with two sinusoidal signals of amplitudes , , and periods , , respectively (which gives a ghost resonance of ). The bias current for all three neurons is , the synaptic coupling between input and processing neurons is and . All noise amplitudes are zero, .

Distribution of inter-spike intervals of the input neurons in two cases: (a) Both neurons with super-threshold modulation ( , ) plus noise ( , ); and (b) both neurons modulated with a sub-threshold harmonic current ( , ) plus optimal noise ( , ), i.e., in the stochastic resonance regime. The bias currents are and .

Distribution of inter-spike intervals of the input neurons in two cases: (a) Both neurons with super-threshold modulation ( , ) plus noise ( , ); and (b) both neurons modulated with a sub-threshold harmonic current ( , ) plus optimal noise ( , ), i.e., in the stochastic resonance regime. The bias currents are and .

Left panels: Response of the processing neuron for increasing noise amplitude: (a) Mean time between spikes , (b) coefficient of variation , and (c) fraction of pulses spaced around , and as a function of the noise amplitude in the processing neuron, . Right panels: Probability distribution functions of the time between spikes for three values of the noise amplitude : (d) , (e) , and (f) . Parameters are and for the synapses and we used , (which gives ). Other parameters are those of Fig. 2(a) , except for the triangles in plot (c), which correspond to Fig. 2(b) .

Left panels: Response of the processing neuron for increasing noise amplitude: (a) Mean time between spikes , (b) coefficient of variation , and (c) fraction of pulses spaced around , and as a function of the noise amplitude in the processing neuron, . Right panels: Probability distribution functions of the time between spikes for three values of the noise amplitude : (d) , (e) , and (f) . Parameters are and for the synapses and we used , (which gives ). Other parameters are those of Fig. 2(a) , except for the triangles in plot (c), which correspond to Fig. 2(b) .

Maximum fraction of pulses at the ghost resonance as a function of the timing mismatch between the input modulating currents relative to the period of the current , . Parameters are the same than in Fig. 3 with .

Maximum fraction of pulses at the ghost resonance as a function of the timing mismatch between the input modulating currents relative to the period of the current , . Parameters are the same than in Fig. 3 with .

Left panels: (a) Mean time between spikes, (b) coefficient of variation, and (c) fraction of pulses around and as a function of . Right panels: Probability distribution functions of the inter-spike intervals for three values of : (d) , (e) , and (f) . The value of is different for each value of , chosen so that the processing neuron is below threshold and does not fire in the absence of noise. In particular, for , and for and . The driving frequencies are and (which gives ). Other parameters are those of Fig. 3(e) (in particular, ).

Left panels: (a) Mean time between spikes, (b) coefficient of variation, and (c) fraction of pulses around and as a function of . Right panels: Probability distribution functions of the inter-spike intervals for three values of : (d) , (e) , and (f) . The value of is different for each value of , chosen so that the processing neuron is below threshold and does not fire in the absence of noise. In particular, for , and for and . The driving frequencies are and (which gives ). Other parameters are those of Fig. 3(e) (in particular, ).

(Color) Left: Fraction of pulses occurring at intervals equal to the period of the ghost resonance . Right: Coefficient of variation . Both quantities plotted as function of noise amplitude and .

(Color) Left: Fraction of pulses occurring at intervals equal to the period of the ghost resonance . Right: Coefficient of variation . Both quantities plotted as function of noise amplitude and .

Probability of observing a spike in the processing neuron with instantaneous rate (in gray scale) as a function of the frequency of one of the input neurons. We can observe a remarkable agreement of the responses following the lines predicted by Eq. (12) for (dashed lines from top to bottom). Parameters: , , , , .

Probability of observing a spike in the processing neuron with instantaneous rate (in gray scale) as a function of the frequency of one of the input neurons. We can observe a remarkable agreement of the responses following the lines predicted by Eq. (12) for (dashed lines from top to bottom). Parameters: , , , , .

## Tables

Parameters values of the Morris–Lecar and synapse models used in this work.

Parameters values of the Morris–Lecar and synapse models used in this work.

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