Characteristic effects of stochastic oscillatory forcing on neural firing: analytical theory and comparison to paddlefish electroreceptor data

Abstract

Stochastic signals with pronounced oscillatory components are frequently encountered in neural systems. Input currents to a neuron in the form of stochastic oscillations could be of exogenous origin, e.g. sensory input or synaptic input from a network rhythm. They shape spike firing statistics in a characteristic way, which we explore theoretically in this report. We consider a perfect integrate-and-fire neuron that is stimulated by a constant base current (to drive regular spontaneous firing), along with Gaussian narrow-band noise (a simple example of stochastic oscillations), and a broadband noise. We derive expressions for the nth-order interval distribution, its variance, and the serial correlation coefficients of the interspike intervals (ISIs) and confirm these analytical results by computer simulations. The theory is then applied to experimental data from electroreceptors of paddlefish, which have two distinct types of internal noisy oscillators, one forcing the other. The theory provides an analytical description of their afferent spiking statistics during spontaneous firing, and replicates a pronounced dependence of ISI serial correlation coefficients on the relative frequency of the driving oscillations, and furthermore allows extraction of certain parameters of the intrinsic oscillators embedded in these electroreceptors.

Figure 1. Illustration of the neuron model, showing a calculated membrane voltage trace (upper panel) that yields a spike time whenever a threshold level is reached, and sample trajectories of input narrow-band harmonic noise (middle panel) and broadband short-correlated Ornstein-Uhlenbeck (OU) noise (lower panel).

Figure 2. Comparison of ISI distributions obtained from numerical simulation or theory Eqs.(6–9) for A and Eqs. (43–46) for B.

A: ISI distributions for different frequency ratios: (top panel), (middle panel), or (bottom panel). Parameters: , , , and . B: Example of a multimodal ISI histogram at high relative driving frequency , with harmonic noise input that was nearly periodic (large quality factor ). Parameters: , , , , , and, consequently, .

Figure 3. Second and third-order interval statistics as a function of the frequency ratio for different values of the OU broadband input (remaining parameters , , and ).

A: Coefficient of variation (CV) for (A1), (A2), or (A3). B: Skewness for (B1) (B2), or (B3). Theoretical CV and skewness (blue) were computed by numerical integration from the theoretical ISI distribution Eq.(6)–Eq.(9) ; the simpler expression Eq.(11) is shown in red in A.

Figure 4. Serial correlation coefficients.

A: SCC value (ranging from −1 to +1) as a function of the lag of interspike intervals, for different values of the frequency ratio: (A1), (A2), (A3), (A4), (A5), or (A6). Parameters: , , and . Dots: simulation. Lines: theory. B: SCC at lag , , for highly coherent harmonic noise, as a function of . Parameters: , , . Theoretical curves were computed from Eq.(11) and Eq.(12).

Figure 5. Correlation lag , in units of the mean interspike interval, as a function of the frequency ratio .

Dots show results of numerical simulations; blue lines show theory according to analytical evaluation of the sum Eq.(14). A: Correlation lag for different values of the quality factor of harmonic noise input: (A1), (A2), and (A3). Parameters: , and . B: Correlation lag at different levels of OU broadband noise: (B1), (B2), and (B3). Parameters: , , and .

Figure 6. Histograms of firing statistics for a sample of paddlefish ER afferents, including distributions of the mean firing rate, , A, the ratio of EO to AO frequencies, , B, and the coefficient of variation, CV, C.

These graphs are for different number of afferents than used in Fig. 6 of Ref. .

Figure 7. Experimental data from three representative paddlefish electroreceptor afferents (dots in A1–A3, dots and error bars in B1–B3, and gray lines in C1–C3), compared to theory, for the values of the frequency ratio, , listed at the top.

A: ISI probability density functions (PDFs). Theoretical PDFs (red, blue, green lines) were calculated using Eq.(6) with ; , 0.2 and 0.5 (legend in A2), and other parameters derived from fitting the SCCs, as explained in the Methods, final section. B: Serial correlation coefficients (SCCs). Theoretical red lines show least square fits using Eq.(12). C–D: Power spectral densities (PSDs). Theoretical lines (magenta) were obtained from numerical simulation of the PIF model using Eqs.(1–4), with , and other parameters the same as for theoretical curves in panels A1–3, derived from the SCC fitting procedure.

Figure 8. Comparison of experimental data with theory for the sample of paddlefish ERs.

A: ISI correlation lags calculated from the experimental data according to Eq.(14), vs. values from theory, Eq.(15), calculated using parameters obtained from fitting experimental SCCs. B: Skewness of experimental vs. theoretical ISI distributions. 45° line is shown by dashed strokes on both panels.

Figure 9. Statistical properties of afferent ISIs (ordinates) versus parameters of epithelial oscillations (abscissas), estimated from experimental data for the sample of n = 56 paddlefish ERs (filled circles).

A,B: The ISI correlation lag characteristic, , versus values of (A), or the frequency ratio (B), of epithelial oscillations. C: Skewness of ISI distributions versus frequency ratio . Blue lines: Theoretical results from PIF models for parameters extracted from each ER by fitting (Methods, final section), while varying or . Solid blue lines: Mean curves for the sample. Dashed blue lines: standard deviation.

Figure 10. Comparison of ISI statistics from numerical simulation and theory versus noise strength for different values of the frequency ratio as indicated in the legends: coefficient of variation (A) with a double logarithmic plot of the same data in the inset, skewness of ISI density (B), and serial correlation coefficient at lag one (C).

Remaining parameters: , , , and .