Mathematical Modeling and Analyses of Interspike-Intervals of Spontaneous Activity in Afferent Neurons of the Zebrafish Lateral Line

Abstract

Without stimuli, hair cells spontaneously release neurotransmitter leading to spontaneous generation of action potentials (spikes) in innervating afferent neurons. We analyzed spontaneous spike patterns recorded from the lateral line of zebrafish and found that distributions of interspike intervals (ISIs) either have an exponential shape or an "L" shape that is characterized by a sharp decay but wide tail. ISI data were fitted to renewal-process models that accounted for the neuron refractory periods and hair-cell synaptic release. Modeling the timing of synaptic release using a mixture of two exponential distributions yielded the best fit for our ISI data. Additionally, lateral line ISIs displayed positive serial correlation and appeared to exhibit switching between faster and slower modes of spike generation. This pattern contrasts with previous findings from the auditory system where ISIs tended to have negative serial correlation due to synaptic depletion. We propose that afferent neuron innervation with multiple and heterogenous hair-cells synapses, each influenced by changes in calcium domains, can serve as a mechanism for the random switching behavior. Overall, our analyses provide evidence of how physiological similarities and differences between synapses and innervation patterns in the auditory, vestibular, and lateral line systems can lead to variations in spontaneous activity.

Figure 1

Comparison of ISI Distribution to Exponential Distributions. (A) Distributions of two ISI data sets in comparison to fitted exponential distributions. (B) Differences between empirical and best-fit exponential CDFs, E_total and the fractional difference by quartiles E_k/E_total. (C) Clustering of data by CDF differences in the first quartile E₁ versus the third quartile E₃. (D) Box plot of mean, CV and kurtosis of ISI data sets with $\ell$ < 1 (in gray) and $\ell$ > 1 (black).

Figure 2

Data-fitting to renewal equations with three different excitation time distributions f_E(t). (A) Empirical CDFs of two ISI data sets (same data sets as in Fig. 1A) in comparison to the theoretical CDF of f_ISI(t) (Eq. (8)) with exponentially-distributed excitation time f_E(t) (case (i)). (B) Differences between empirical and best-fit CDFs (as defined in Eq. (6)) for case (i) by quartiles for datasets ($\ell$ < 1 in gray and $\ell$ > 1 in black). (C) Empirical CDFs of two ISI data sets in comparison to theoretical CDF with gamma-exponential mixture excitation time (case (ii), same data sets as in (A)). (D) Best-fit values for 1 − p, the fraction events generated by the gamma distribution, against n, the gamma shape parameter. Dot size corresponds proportionally to the total difference between the empirical CDF and model CDF (larger dots correspond to larger differences). (E) Empirical CDFs of two ISI data sets in comparison to renewal CDF with two exponentials mixture excitation time (case (iii), same data sets as in (A)). (F) Box plots of best-fit rate parameters values λ_E1 and λ_E2 for datasets ($\ell$ < 1 in gray and $\ell$ > 1 in black). (G) Comparisons of total difference in empirical and theoretical CDFs (as defined in Eq. (6)) for for the three different models considered (case (i–iii)).

Figure 3

Dependencies between sequential ISIs. (A) Box plot of SRC for consecutive ISI pairs (SRC(1)) for original unsorted data sets and shuffled data set. (B) Visualizations of the renewal quartile matrix q_ij showing frequencies for an ISI at a given quartile (horizontal axis) to be followed by an ISI at another quartile (vertical axis). Three qualitatively different mappings can be seen for data sets with positive correlation, zero correlation and negative correlations. (C) Boxplot of entries of renewal quartile matrices across all data sets. Compared to an equal likelihood for any ISI to fall in any quartile (q_ij = 1/16 shown as dashed line), we saw overall increases in q₁₁ (short followed by short) and q₄₄ (long-long), while q₁₄ (short-long) and q₄₁ (long-short) were reduced. (D) Groupings of long and short ISIs that occur consecutively in sequence over time. (E) Fraction of ISIs that are found within long and short sequences for data sets with negative, zero (uncorrelated), and positive SRC(1) (each dot represent a dataset).

Figure 4

Dependencies of serial correlations for the two-state switching model on parameter values. (A) Box plots of SRC(1) values measuring correlations between consecutive ISIs for different values of switching rate k_sf (k_fs is varied so that p_fast is constant at 0.6, k_fs = k_sf (1−p_fast)/p_fast. All other parameter values were fixed: p_fast = 0.6, τ_fast = 40 ms, τ_slow = 200 ms, t_abs = 2 ms, and t_rel = 2 ms). (B) Box plots of SRC(1) values for different ratios of τ_fast/τ_slow (all other parameter values were fixed: k_sf = 10⁻⁴/ms, p_fast = 0.6, τ_fast = 40 ms, t_abs = 2 ms, and t_rel = 2 ms). (C) Box plots of SRC(1) values for different values of p_fast using two different values of switching rates k_sf = 10⁻³/ms in black and k_sf = 10⁻⁴/ms in gray (all other parameter values were fixed: τ_fast = 40 ms, τ_slow = 200 ms, t_abs = 2 ms, and t_rel = 2 ms). For each box plot in this figure, 100 trials were performed with each trial consisting of 5000 ISIs.

Figure 5

Long term dependencies between consecutive spikes in lateral-line data and the two-state switching model. (A) Fano factor as a function of counting time for the lateral line data (each curve corresponded to a dataset). (B) Fano factor obtained from simulating the two-state switching model using two different synaptic release time ratios: top: τ_fast = 40 ms, and τ_slow = 80/ms and bottom: τ_fast = 40 ms, and τ_slow = 160/ms. (All other parameter values were fixed: k_sf = 10⁻⁴/ms, p_fast = 0.4, k_fs = k_sf (1 − p_fast)/p_fast, t_abs = 2 ms, and t_rel = 2 ms). (C) SRC(n) at different lag time n (see definition in Eqn 9) for the lateral line data. (D) SRC(n) at different lag time n obtained from simulating the two-state switching model using two different values of switching rates k_sf = 10⁻³/ms in black and k_sf = 10⁻⁴/ms in gray (all other parameter values were fixed: τ_fast = 40 ms, p_fast = 0.4, τ_slow = 200 ms, t_abs = 2 ms, and t_rel = 2 ms). Each dot corresponded to the mean SRC(n) value from simulating 100 trials with each trial consisting of 2000 ISIs).

Figure 6

Simulation results from the depletion-replenishment model. (A) Comparison of ISI distribution generated from the model under synaptic depletion (τ_repl = 2.5 ms, p_depl = 0.08/ms, n_max = 4, 10,000 ISIs were generated) in comparison to the best-fit exponential distribution (analogous to Fig. 1A). Due to synaptic depletion, shorter ISI becomes less likely ($\ell =0.0352$). (B) Colormap of SRC(1) values obtained by random sampling of values for τ_repl and p_depl (n_max was fixed at 4). Each dot corresponded to a trial with 2000 ISIs generated; a total of 2000 dots/trials were generated. (C) The effects of varying the number of sources or hair cells on SRC(1) values. Each boxplot consisted of 100 trials where 2000 ISIs were generated per trial (τ_repl = 2.5 ms, p_depl = 0.08/ms and n_max = 4 for all sources). (D) Colormap of SRC(1) values obtained when parameters associated with the two different sources were allowed to vary from one another. Ratios of p_depl and τ_repl between the two sources are used as axis values. Each dot corresponded to a trial with 2000 ISIs generated; a total of 4440 dots/trials are shown. (E) Top: Shape of ISI distribution obtained with two sources with parameter values chosen to give rise to SRC(1) > 0. Bottom: Shape of ISI distributions if only the first or second source generated spikes in the absence of the other (${\tau }_{repl}^1=20.90$ ms, $p_{depl}^1=0.0028$/ms, ${\tau }_{repl}^2=90.30$ ms, $p_{depl}^2=0.715$/ms $n_{max}^1=n_{max}^2=4$; 2000 spikes were generated for each histogram).