Emergence of spike correlations in periodically forced excitable systems

In sensory neurons the presence of noise can facilitate the detection of weak information-carrying signals, which are encoded and transmitted via correlated sequences of spikes. Here we investigate the relative temporal order in spike sequences induced by a subthreshold periodic input in the presence of white Gaussian noise. To simulate the spikes, we use the FitzHugh-Nagumo model and to investigate the output sequence of interspike intervals (ISIs), we use the symbolic method of ordinal analysis. We find different types of relative temporal order in the form of preferred ordinal patterns that depend on both the strength of the noise and the period of the input signal. We also demonstrate a resonancelike behavior, as certain periods and noise levels enhance temporal ordering in the ISI sequence, maximizing the probability of the preferred patterns. Our findings could be relevant for understanding the mechanisms underlying temporal coding, by which single sensory neurons represent in spike sequences the information about weak periodic stimuli.

Figure 1

Time series generated from the FHN model with the parameters $a=1.05$, $\epsilon =0.01$, and $D=0.015$. In (a) $a_0=0$, while in (b) and (c) $a_0=0.02$ and $T=10$ and 20, respectively. The spike times are detected with the threshold $y=1.5$. In (b) and (c) the dashed line indicates the value of $cos(2\pi t/T)$.

Figure 2

Probabilities of the six OPs that are defined by the relative length of three consecutive ISIs vs the noise strength. The OPs are schematically shown in the inset. The parameters are (a) $a_0=0$, (b) $a_0=0.02$ and $T=10$, and (c) $a_0=0.02$ and $T=20$; the other parameters are as indicated in the text. In (c) the arrows indicate the noise levels used in Figs. 3 and 4.

Figure 3

The OP probabilities vs the amplitude of the input signal. The parameters are $T=20$ and (a) $D=0.015$ and (b) $D=0.035$; the other parameters are the same as in Fig. 1.

Figure 4

The OP probabilities vs the period of the input signal. The parameters are $a_0=0.02$ and (a) $D=0.015$ and (b) $D=0.035$; the other parameters are the same as in Fig. 1.

Figure 5

(a) Probabilities of patterns 01 and 10 vs the period of the input signal. (b) Permutation entropy vs $T$ for OPs of length $L=3$, 4, and 5. In both (a) and (b) the parameters are the same as in Fig. 4.

Figure 6

(a) Probabilities of patterns 01 and 10 vs the number $M$ of interspike intervals in logarithmic scale. (b) Probabilities of the six $L=3$ OPs vs $M$. For both (a) and (b) the parameters are $a_0=0.025$, $D=0.015$, and $T=20$. (c) Same as in (b) but with $a_0=0$.

Figure 7

(a)–(c) OP probabilities, (d)–(f) serial correlation coefficients ($C_1$ circles, $C_2$ triangles), and (g)–(i) mean interspike interval. The parameters are the same as in Fig. 2: (a) $a_0=0$, (b) $a_0=0.02$ and $T=10$, and (c) $a_0=0.02$ and $T=20$.

Figure 8

(a) and (b) OP probabilities, (c) and (d) serial correlation coefficients ($C_1$ circles, $C_2$ triangles), and (e) and (f) mean interspike interval. The parameters are the same as in Fig. 3: $T=20$ and (a) $D=0.015$ and (b) $D=0.035$.

Figure 9

(a) and (b) OP probabilities, (c) and (d) serial correlation coefficients ($C_1$ circles, $C_2$ triangles), and (e) and (f) mean interspike interval. The parameters are the same as in Fig. 4: $a_0=0.02$ and (a) $D=0.015$ and (b) $D=0.035$.

Figure 10

Scatter plots in which all the data sets shown in the three previous figures are collapsed. Here the OP probabilities are plotted vs (a), (c), and (e) $C_1$ and (b), (d), and (f) $C_2$. For clarity the OP probabilities are separated in three groups: the trend patterns (a) and (b) 012 and 210 and the two clusters of patterns that have similar probabilities (c) and (d) 021 and 102 and (e) and (f) 120 and 201. No clear relation between $C_1$, $C_2$, and the six ordinal probabilities is seen, but there is a well-defined trend with $C_2$.

Figure 11

Scatter plots of ordinal probabilities vs $C_1$ and $C_2$. The color code indicates the value of (a) $P\left(012\right)$ and (b) $P\left(102\right)$. Both panels display the same data that were shown in Figs. 7–9.