A model neuron with activity-dependent conductances regulated by multiple calcium sensors

Abstract

Membrane channels are subject to a wide variety of regulatory mechanisms that can be affected by activity. We present a model of a stomatogastric ganglion (STG) neuron in which several Ca2+-dependent pathways are used to regulate the maximal conductances of membrane currents in an activity-dependent manner. Unlike previous models of this type, the regulation and modification of maximal conductances by electrical activity is unconstrained. The model has seven voltage-dependent membrane currents and uses three Ca2+ sensors acting on different time scales. Starting from random initial conditions over a given range, the model sets the maximal conductances for its active membrane currents to values that produce a predefined target pattern of activity approximately 90% of the time. In these models, the same pattern of electrical activity can be produced by a range of maximal conductances, and this range is compared with voltage-clamp data from the lateral pyloric neuron of the STG. If the electrical activity of the model neuron is perturbed, the maximal conductances adjust to restore the original pattern of activity. When the perturbation is removed, the activity pattern is again restored after a transient adjustment period, but the conductances may not return to their initial values. The model suggests that neurons may regulate their conductances to maintain fixed patterns of electrical activity, rather than fixed maximal conductances, and that the regulation process requires feedback systems capable of reacting to changes of electrical activity on a number of different time scales.

Fig. 1.

Three different activity patterns have similar average [Ca²⁺] levels. A, Left, Membrane potential of a neuron firing action potentials tonically.Right, The instantaneous Ca²⁺concentration (oscillating curve) and its time-averaged value (approximately straight line). The average [Ca²⁺] level is 4.3 μm. B, Left, Membrane potential for a neuron firing bursts of action potentials. Right, [Ca²⁺] and its average value. Average [Ca²⁺] level is 4.0 μM.C, Left, Membrane potential for a neuron firing in a different bursting pattern. Right, [Ca²⁺] and its average value. Average [Ca²⁺] level is 4.3 μm.

Fig. 2.

The three Ca²⁺ sensors distinguish different activity patterns. Rows A–Ccorrespond to the same three patterns of activity presented in Figure1, as can be seen by the membrane potential plots in the first column. The second column shows the Ca²⁺ current in each case, and the remaining three columns show the transient (oscillating curves) and average values (approximately straight lines) of the fast, slow and DC Ca²⁺sensors F, S, and D. Note that, taken collectively, the average values now distinguish among the three different types of activity.

Fig. 3.

Approach to equilibrium from two different initial conditions. The top traces in A andB show the activity of the model neuron with two randomly chosen sets of initial maximal conductance values. Over time, the model dynamically adjusted the maximal conductances of the seven active currents of the model until the activity shown in the lower traces was obtained. The values of the maximal conductances as a function of time over 15 sec is shown for both cases in thebottom row of plots. The vertical axes forCaT, CaS, and H extend from 0 to 2 μS/nF whereas the range is 0–50 μS/nF for all other conductances. In both A and B, the model achieves a bursting pattern of activity but the final equilibrium values of the maximal conductances are different (bottom plot).

Fig. 4.

Range of equilibrium conductance values for a bursting model neuron. The model was run repeatedly starting from randomly chosen initial maximal conductance values until a steady-state pattern of bursting activity was attained. The initial maximal conductances for CaT, CaS, andH were chosen uniformly in the range between 0.05 and 0.95 μS/nF, whereas the maximal conductances for the remaining active currents were chosen randomly between 2.5 and 47.5 μS/nF. Thepoints show final steady-state maximal conductances for 31 runs. A, Range of steady-state maximal conductances. Note that the maximal conductances for some of the currents have been multiplied by 10 to make them more visible. B, Maximal conductances of the three outward currents in each run plotted against each other to show that no strong correlation or pattern emerges.

Fig. 5.

Range of conductance densities for K⁺ currents measured in 12 LP neurons from the crab STG. Each point represents a different neuron.A, Distribution of conductance densities measured.B, Conductance densities for the three K⁺ currents in individual neurons plotted against each other. As in Figure 4B, no obvious correlation or pattern can be seen.

Fig. 6.

Response to an external perturbation.A, The model was at equilibrium producing the bursting activity shown. B, The membrane potential immediately after the reversal potential for the K⁺ currents was changed from −80 mV to −60 mV. C, The activity of the model after a new equilibrium configuration of maximal conductances developed in response to the perturbation. D, The membrane potential immediately after the K⁺ reversal potential was set back to −80 mV. E, Recovery of the model back to the initial bursting activity. The plots at theright show the maximal conductances corresponding to these different cases. These are not shown for B andD, because they are identical, respectively, to the histograms in A and C. Note the increase in Na and K_d conductances inC and that the conductances in A andE are not identical.

Fig. 7.

Effect of removing theI_H conductance. A, Initial activity and maximal conductances of the model at equilibrium.B, Activity of the model immediately after theI_H conductance was set to 0. The conductance histogram at right is identical to that ofA, except that ḡ_H = 0.C, The new equilibrium activity and conductances established by the model.

Fig. 8.

Simulation of experiments done on cultured STG neurons (Turrigiano et al., 1994). A, The activity of the model neuron in its initial equilibrium configuration.B, Activity during a series of hyperpolarizing current pulses applied to the model. The injected current is plotted below the membrane potential trajectory. C, Same asB but after more prolonged exposure to hyperpolarizing current pulses. D, The activity of the model immediately after the prolonged sequence of hyperpolarizing pulses was terminated.E, Activity somewhat longer after the hyperpolarizing pulses were terminated. F, Recovery of the model back to its initial state.

Fig. 9.

Range of steady-state activities obtained using different target values for the slow and fast Ca²⁺sensors. In all cases D = 0.1. Other values were: A, F = 0.25, S = 0.09; B,F = 0.2,S = 0.09; C,F = 0.06,S = 0.09; D,F = 0.15,S = 0.045; E,F = 0.2,S = 0.045; F,F = 0.06,S = 0.045.

Fig. 10.

Parameters and functions used to describe the membrane currents of the model. Notation is explained in the text. All membrane potentials are in millivolts, time constants are in milliseconds, and [Ca] refers to the micromolar intracellular Ca²⁺ concentration.