Table 1.
IF model default parameters.
Table 2.
IF model: Genetic algorithm fit parameters.
Fig 1.
Reference data for model fitting.
(A) shows the mean firing rate of an oxytocin neuron (in 16-s bins) recorded from the supraoptic nucleus of a urethane-anesthetised rat. The neuron was antidromically identified as projecting to the posterior pituitary gland and identified as an oxytocin neuron by the transient excitation in response to CCK (arrow). The neuron was recorded throughout an i.v. infusion of hypertonic saline (yellow bar) which increased its firing rate from 2.3 spikes/s to 9.0 spikes/s. See [9] for full experimental details. (B) shows on the left, the ISI distribution for the 850-s period of low frequency activity (1945 spikes) indicated by the light blue background, and on the right, the ISI distribution for the 2000-s period of high frequency activity (17236 spikes) indicated by the dark blue background (S1 Data). These distributions are typical of oxytocin cells, and have been normalised to 10000 ISIs. The ISI distributions after the mode are well fit by a negative exponential (black dotted lines, equations of best fit given). Fewer spikes produces a noisier distribution. (C) shows the hazard functions (in the same 5-ms bins) corresponding to the ISI distributions in B. A flat hazard indicates no time dependent influence on the chance of firing. The initial climb from 0 shows the effect of the HAP.
Fig 2.
Adding synaptic input to the oxytocin neuron HH model.
(A) The original oxytocin HH model, challenged with a constant applied current of 4.5 nA produces regular firing at 10 spikes/s; each spike is followed by a conspicuous but short HAP. (B) shows the HH model with EPSCs at 600 Hz (Repsc = 600, Ripsc = 0), generating spikes at a mean rate of 5.6 spikes/s. The randomly timed EPSCs generate a noisy spiking membrane potential that much more closely resembles an in vivo neuron than the regular spikes produced by an applied current. Note that the post-spike HAP is less conspicuous and of more varied appearance than in A. (C) shows the HH model with EPSCs at 600 Hz, and IPSCs at 300 Hz (Repsc = 600, Ripsc = 300), generating spikes at a mean rate of 1.6 spikes/s. Note that the HAP following spikes is even less conspicuous than in B.
Table 3.
HH model synaptic input parameters.
Fig 3.
Fitting the IF model using a genetic algorithm.
A genetic algorithm was used to find values for five parameters of the IF model that give an optimal fit to the reference data of Fig 1. (A) shows how a single run of the algorithm (128 parameter sets or ‘chromes’) converges over 20 generations. The left panel shows the five parameters, plotting the coefficient of variation (CV) for each parameter across the population. The right panel shows mean population fitness improving. The best fit parameters and fitness score (range 3.69 to 7.38) varied between runs. (B) shows that the values found for the half-life of the HAP (λHAP) are inversely related to the values found for the magnitude of the HAP (kHAP), and that the values found for the half-life of the AHP (λAHP) are inversely related to the values found for the magnitude of the AHP (kAHP): thus these parameters are not independent, but compensate against each other to some extent. (C) plots found parameters against relevant elements of the fit measure. The plots in C and D are colour coded by overall fit measure. Red shows the top 25%, green 25% to 50%, and blue the bottom 25%. The outliers tend to have poorer fit scores, whereas the red values are mostly clustered in a small range. To choose a single best fit we took the median value for parameters from the 10 best fits, shown by the white dots, which each fall within the red clusters. The final parameters (for the 9 spikes/s data) are Ire = 648 Hz, kHAP = 83 mV, λHAP = 8 ms, kAHP = 0.77, and λAHP = 482 ms.
Fig 4.
The IF model genetic algorithm fitted to in vivo oxytocin data.
(A) shows in blue the ISI distribution in the reference data (Fig 1) for spike activity at 2.3 spikes/s (left) and 9 spikes/s (right), and in green the fit achieved with the parameter set found by the genetic algorithm (Fig 3). We used the parameters given in Fig 3 to generate the fit to the data at 9 spikes/s; to fit the 2.3 spikes/s data we reran the genetic algorithm with all parameters fixed except the synaptic input rate (Ire). This achieved a good fit with Ire = 334 Hz. (B) shows the corresponding matches to IoD values (reference data in blue, model data in green).
Fig 5.
Fitting the HH model to ISI distributions.
(A) shows examples of the HH model spiking activity used to generate the ISI distributions. (B) shows in maroon the ISI distributions generated by the original HH model, at firing rates of 2.3 spikes/s (left) and 9 spikes/s (right), compared with reference data (blue). These were achieved with EPSC rates (Repsc) of 505 Hz and 769 Hz. In both the proportion of short ISIs is larger than in the reference data (in blue), and the mode is earlier. (C) shows the ISI distributions from the HH model with τpd, the deactivation time constant of IBK, increased from 1.22 to 10 ms. EPSC rates (Repsc) were slightly increased, to 506 Hz and 774 Hz, to reacquire the firing rates of 2.3 and 9 spikes/s. In the adjusted model, the ISI distributions fit the reference distributions well at both firing rates.
Fig 6.
Spike waveforms in the HH model with varied BK.
(A) shows the voltage, IBK current, and IBK activation behaviour of a spike induced by a 5-nA current step for 5 ms, in the original oxytocin HH model (yellow, τpd = 1.22 ms), and the adjusted model (green; τpd = 10 ms). Experimentally, the HAP lasts for 25–125 ms and hyperpolarises the cell by ~7.5 mV [18]. In the original HH model, the HAP is too short and too small, because IBK both rises and falls sharply after a spike. In the adjusted model, IBK is prolonged by the slower deactivation, making a better match to the HAP observed in vitro, as well as improving the fit to the in vivo ISI distribution. (B) shows, for the adjusted model, the effects of setting gBK = 0 (red, compared to original value gBK = 1, in green), on a single spike induced by an 8-nA current step for 3 ms, to simulate the pharmacological block of IBK [18]. When IBK is blocked, the HAP is smaller and decays faster. (C) shows the in vitro experiment data simulated in (B), redrawn from [18]. The duration and amplitude of the HAP is a much closer match to the adjusted model. The block shows only a partial reduction, but is likely to be less total than the model.
Fig 7.
Changes in EPSC parameters have little effect on the ISI distribution.
We investigated how changes to the EPSC parameters affect the ISI distribution in the HH model when the spike rate is kept constant at 9 spikes/s. (A) Increasing the EPSC magnitude (Δepsc) slightly increases the number of short ISIs, shifting the distribution to the left, with a small decrease in the height of the mode. Plotted distributions are for EPSP magnitudes of ~ 1, 2, 3, and 4 mV (darkest to lightest). The changes in EPSPs that result from the changes in kepsc are plotted on the right. EPSC rates (Repsc = 2920, 1294, 774 and 526 Hz respectively) were adjusted to achieve a firing rate of 9 spikes/s. (B) Increasing the EPSC time-constant (τepsc) very slightly increases the number of short ISIs, shifting the distribution to the left, with a small decrease in the height of the mode. Plotted distributions are for τepsc = 1.1, 2.2, 3.3, and 4.4 ms (yellow to dark red). The changes in EPSPs that result from the changes in τepsc are plotted on the right. EPSC rates (Repsc = 2685, 1220, 774, and 567 Hz respectively) were adjusted to achieve a firing rate of 9 spikes/s.
Fig 8.
Effects of adding inhibitory synaptic input.
(A) shows the effect of increasing Iratio to 0.5 from 0 on the ISI distributions generated by the HH model at 2.3 and 9 spikes/s (Repsc was adjusted to 694 and 1304 Hz to achieve firing rates of 2.3 and 9 spikes). Model distributions (pink; Iratio = 0.5, gBK = 1) are superimposed on ISI distributions from the reference data in Fig 1 (blue). Increasing Iratio increases the number of short ISIs, and the model ISI distribution no longer matches the reference data. (B) shows the effect of increasing gBK to 3.2 mS/cm2, with Repsc adjusted to 694 and 1290 Hz to achieve firing rates of 2.3 and 9 spikes/s. ISI distributions from the adjusted model (purple; Iratio = 0.5, gBK = 3.2) are superimposed on ISI distributions from the reference data in Fig 1 (blue). By increasing gBK, a good fit can again be obtained. (C) shows the ‘average spike waveform’ for three different runs of the model–Iratio = 0 in green (from Fig 6B), Iratio = 0.5, gBK = 1 in pink, and Iratio = 0.5, gBK = 3.2 in purple. Increasing Iratio results in an attenuation of the observed HAP, which can be reversed by increasing gBK, restoring a good fit to the in vivo reference data.
Fig 9.
Fitting the HH model to the IoD range.
The IoD measured at a range of selected binwidths shows the timescale dependence of spike rate variability. (A) compares the IoD range of the reference data, at 2.3 (light blue) and 9 spikes/s (light blue), with two variations of the HH model. Data from the model of Fig 8 are in purple. The data in red are from the model with stronger Ca2+ activation of the AHP generating SK channel (αSK changed from 1.6 to 4.5). The EPSC rate (Repsc) was increased to 746 and 1860 Hz to achieve firing rates of 2.3 and 9 spikes/s with Iratio = 0.5. The previous (Fig 8) version of the HH produces IoD values that are too high, implying that spike events are more variable than found experimentally. In the model with αSK increased to 4.5, the IoD values closely match the reference data at both spike rates, suggesting that the AHP plays a large role in spike rate variability in an oxytocin cell. (B) compares the ISI distributions of the model with αSK = 4.5 to the reference data. With the changes to the AHP, the model ISI distributions still fit experimental data well at both firing rates. Thus, for a given firing rate, the parameters of the AHP current have little effect on the ISI distribution.
Table 4.
Adjusted HH model parameters.
Fig 10.
Spike rate response with varied Iratio and AHP.
The panels show how the spiking rate increases with the synaptic input rate (increasing REPSC with varied Iratio = 0, 0.25, 0.5 and 0.75), comparing the HH model (left) and IF model (right). (A) With no AHP (gSK = 0), the HH model with only excitatory input (Iratio = 0) shows a highly non-linear response, which becomes increasingly linear as Iratio is increased. (B) shows a similar result with the genetic algorithm-fitted IF model with kAHP = 0, matching [9]. (C) With the AHP (gSK = 0.18) the HH model shows a more linear response even with Iratio = 0. (D) Similarly in the IF model, the AHP linearises the response to increasing input. Thus, the AHP, in both the HH and IF models, has a similar linearising effect to increasing the ratio of inhibitory input.
Fig 11.
Matching ISI distributions from oxytocin neurons in vitro.
Neurons recorded in vitro (redrawn from [6]), stimulated by a constant depolarising current, show a much narrower ISI distribution, indicating a more regular spiking rate, much closer to a normal distribution than observed in vivo (Fig 1). The remaining variability can be attributed to membrane noise, or some low level of residual synaptic input. Using the same HH and IF models, with parameters fitted to the in vivo reference data, we were able to match the in vitro ISI distribution using a constant applied input combined with a low rate of low magnitude excitatory synaptic input. In the HH model (red) we used a 4.5 nA constant input with EPSC parameters Δepsc = 0.008 and Repsc = 120 Hz, producing 5.3 spikes/s. In the IF model we used Vext = 20.3 mV with EPSP parameters Ire = 120 Hz and eh = 0.08 mV, and Iratio = 0. The cluster plots indicate that ISIs are independent of the preceding ISI.