Fig 1.
Circuit and single cell electrical properties of electrosensory lateral line lobe (ELL) pyramidal cells.
(A) Schematic of the neural circuitry underlying the electrosensory processing in the weakly electric fish (left). Brain circuitry (middle) showing how the ELL receives feedforward input from electrosensory afferents (EAs) and feedback signals from higher brain areas, including the eminentia granularis posterior (EGp), nucleus praeminentialis (nP), and torus semicircularis (TS). Intracellular recordings were obtained from pyramidal neurons within the ELL to investigate their electrical activity (right). (B) Intracellular membrane potential trace recording for 5 sec (left), phase plane plot of action potential cycles in the voltage trace (middle), and interspike interval (ISI) distribution (right) for a representative recording.
Fig 2.
Comparison of electrophysiological features between intracellular recording of a given ELL pyramidal cell and a model simulation.
(A) Schematic of the model showcasing the two-compartments of the cell: soma and dendrite (left), and highlighting the various ionic currents expressed in each compartment, including fast Na + (), delayed rectifier K + (
), small-conductance K + (
), NMDA (
) and leak (
) currents in the soma (i = S) and dendrite (i = D). The kinetics of NMDA receptors are governed by a Markov model (see Methods). The schematic of Ca2 + dynamics within the dendritic compartment following the flux-balance model is also shown (right). It displays all fluxes involved in regulating Ca2 + mobilization across plasma and ER membranes in the dendritic compartment. These fluxes include Ca2 + entry through NMDA receptors (
) and IP3 receptors (
), Ca2 + efflux through SERCA (
) and PMCA (
) pumps, and leak across both membranes. (B) Schematic of an action potential with the extracted electrophysiological features (highlighted in pink) for model fitting; that includes from left to right: spike threshold (threshold), midpoint of the upstroke phase (upstrokev), peak amplitudes (peakv), midpoint of the downstroke phase (downstrokev), trough amplitudes (troughv) and amplitude of afterdepolarization potential (adpv). (C) Extracted action potentials (left) and action potential cycles (right) obtained from one recorded trace (5 sec; shaded dark blue) and the average action potential and average action potential cycle of all spikes obtained from a single model simulation overlaid on top (5 sec; light blue). (D) Box plots of electrophysiological features obtained from all action potentials extracted from the experimental recording and model simulation in C. Panels from left to right: thresholdv, upstrokev, peakv, downstrokev, troughv and adpv. Statistical comparisons were performed using two-sample t-tests, Mann–Whitney U tests, and Kolmogorov–Smirnov tests. No significant differences were found for any feature (thresholdv:
,
,
; upstrokev:
,
,
; peakv:
,
,
; downstrokev:
,
,
; troughv:
,
,
; adpv:
,
,
). For adpv estimation, spikes without a detected ADP were omitted.
Fig 3.
Comparison of spike train features between intracellular recordings of a population of ELL pyramidal cells and their fitted model simulations.
(A) Membrane potential traces (top) and interspike interval (ISI) distribution (bottom) of a representative ELL pyramidal cell (dark blue, left) and its fitted model simulations (light blue, right). The recorded and simulated traces are 5 seconds long, showing sequences of bursts interspersed with isolated spikes and quiescent periods. Insets show zoomed-in examples of individual bursts from the recorded and simulated traces, highlighting the close match in spike shape and temporal organization within bursts. (B) Raster spike train plots from same recording in A (left) and their corresponding fitted simulations (right), separated into isolated spikes (green), burst spikes (red), and all spikes combined (blue). The model replicates the proportions and temporal organization of burst and isolated spikes observed in the recordings. (C) Scatter plots comparing key spike train features between recordings and simulations for all ELL pyramidal cells (n = 32). Panels from left to right: mean firing rate (meanfr; r = 0.99, ), burst fraction (burst frac; r = 0.88,
), mean (meanisi; r = 0.95,
), median (medianisi; r = 0.92,
), standard deviation (stdisi; r = 0.83,
) and the coefficient of variation (CVisi; r = 0.98,
) of ISIs, respectively.
Fig 4.
Bifurcation analysis of the deterministic biophysical model reveals the full dynamical behavior of ELL pyramidal cells.
(A) One-parameter bifurcation diagram of the voltage variable (V) of the biophysical model with respect to the applied current (), illustrating the various regimes of behavior corresponding to branches of stable equilibria (orange line), branches of unstable equilibria (dashed black line), a branch of stable periodic orbits (green lines) and a branch of stable bursting orbits (dashed green line). Within the range of
considered, the model undergoes several types of bifurcation, including 2 saddle-node bifurcations (SN1, SN2), 1 Hopf bifurcation (HB), 2 saddle-node bifurcations of periodic orbits (SNP1, SNP2) and 2 period-doubling bifurcations (PD1, PD2), some of which define the boundaries of the various regimes of behavior identified. The inset provides magnified views of the area enclosed by the highlighted region in the main figure. (B) The two-parameter bifurcation diagram of the biophysical model with respect to
and the maximum conductance of SK channels (
), showing the distinct dynamical regimes of the model, including quiescence, tonic firing and doublet/chaotic ghostbursting. The region bounded by PD1 is multistable, featuring coexisting distinct attractors. (C) The two-parameter bifurcation diagram of the biophysical model with respect to
and the maximum conductance of NMDA receptors (
), showing the distinct dynamical regimes of the model, including quiescence, tonic firing and doublet/chaotic ghostbursting. (D) A sample of three principal firing patterns: tonic spiking at high (1), and low frequencies (2), ghostbursting in the form of doublets (3) and chaotic ghostbursting (4) at different values of
. These patterns demonstrate the rich repertoire of activity captured by the biophysical model.
Fig 5.
Bifurcation analysis of the deterministic modified Hindmarsh-Rose (HR) model shows similar dynamic regimes as those of the full model.
(A) One-parameter bifurcation diagram of the voltage variable (V) of the modified HR model with respect to the applied current (), illustrating the various regimes of behavior corresponding to branches of stable equilibria (orange line), branches of unstable equilibria (dashed black line), branches of stable periodic orbits (green lines), branches of bursting orbits (dashed green line), and branches of unstable periodic orbits (dashed blue line). Within the range of
considered, the model undergoes several types of bifurcation points, including 2 saddle-node bifurcations (SN1, SN2), 6 Hopf bifurcations (HB1-HB6), 2 saddle-node bifurcations of periodic orbits (SNP1, SNP2), 2 homoclinic bifurcations (HM1, HM2), 2 period-doubling bifurcations (PD1, PD2) and 1 torus bifurcation (TS), some of which define the boundaries of the various regimes of behavior identified. The two insets provide magnified views of the areas enclosed by the bounding boxes in the main figure: the top inset corresponds to the larger, more physiologically relevant box, while the bottom one corresponds to the smaller, unphysiological box. (B) Plot of the firing rate of the modified HR model over a 20 sec interval with respect to
, highlighting chaotic dynamics consistent with the biophysical model (compare to S4B Fig). (C) A sample of three principal firing patterns: tonic spiking (
, left), ghostbursting in the form of doublets (
, middle) and chaotic ghostbursting (
, right). These patterns demonstrate the rich repertoire of activity captured by the modified HR model.
Fig 6.
Impact of stochastic synaptic input on the deterministic dynamics of the modified Hindmarsh-Rose (HR) model.
(A) Simulation of the voltage variable (V) of the modified HR model (top, teal), along with the slow adaptation variables z (red) and u (yellow), for an example ELL pyramidal cell receiving stochastic synaptic input (see Methods). (B) Raster spike trains obtained from simulations of the modified HR model for five different representing cells receiving stochastic synaptic input. These spike trains demonstrate the diversity of firing patterns across cells. (C) Statistical comparisons between experimental data (dark blue) and model simulations (teal) for spike train features of all recorded ELL pyramidal cells. From left to right: mean firing rate (meanfr; r = 0.97, ), burst fraction (burst frac; r = 0.86,
), mean (meanisi; r = 0.91,
), median (medianisi; r = 0.97,
), standard deviation (stdisi; r = 0.77,
) and the coefficient of variation (CVisi; r = 0.98,
) of ISIs, respectively.
Fig 7.
Parameter sensitivity analysis of the modified Hindmarsh-Rose (HR) model reveals significant contribution of the slow adaptation variable and stochastic background noise in aligning it with in vivo spiking activity of ELL pyramidal cells.
(A) Sobol sensitivity indices for the top 10 model parameters in the modified HR model receiving stochastic synaptic input. The violin plots represent the distributions of parameter values, scaled by their importance as determined by the Sobol sensitivity indices for each parameter. The most sensitive parameters, based on higher Sobol indices, are and
, indicating the greater influence of the slow adaptation variable on model behavior compared to other parameters. (B) Comparison of the interspike interval (ISI) distributions between an example recorded ELL pyramidal cell (blue) and ISIs from model simulations without (orange, top) and with (teal, bottom) stochastic synaptic input, highlighting the significantly improved alignment of the model with experimental data in the presence of stochastic synaptic input (Kolmogorov-Smirnov test vs data:
,
;
,
).