Firing patterns in the adaptive exponential integrate-and-fire model

Abstract

For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulations.

Fig. 1

Phase plane representation of a step current injected in an AdEx model where a a saddle-node bifurcation is responsible for the loss of stability, and b the Andronov-Hopf bifurcation is responsible for the loss of stability. In the phase planes the trajectories of the first and second spikes are represented by blue squares and the state of rest is indicated by the blue cross. As the current increases, the V-nullcline shifts upwards. This makes the two fixed pointsmove toward each other. a In the saddlenode bifurcation, the fixed points disappear after the stable fixed point merges with the unstable fixed point. The point where the two fixed points merge lies close to (but slightly to the right of) the voltage V _T , i.e. the minimum of the V-nullcline. b As the stable fixed point moves towards the right, the slope of the V -nullcline increases at the fixed point. If the slope of the w-nullcline is sufficiently high, this can lead to a loss of stability of the stable fixed point before the fixed points disappear. The w-nullcline is shown in green, the V-nullcline in the absence of current is the curved dash line, the V-nullcline in the presence of stimulating current is the curved solid line (black). Unstable fixed points are encircled. The scale bars corresponds to 20 mV vertically and 20 ms horizontally

Fig. 2

The Type of f-I curve depends on the point of reset. a When a = 0 only type I f-I curves are possible. The phase plane at a current just above rheobase (I = I _SN + 0.5 pA) shows that the trajectories are forced to pass through the ghost of the saddle-node bifurcation independently of the starting point (three steady-state trajectories are shown for three different values of V _r voltage reset). The frequency-current plot on the right shows the f-I curve corresponding the the three different reset conditions (consistent line types; full, dash or dot/dash). b When a > 0 it is possible to have type I and type II depending on the reset. The region in pink shows the basin of attraction of the stable fixed point just before (I = I _SN − 0.1 pA) the system loses stability via a saddle-node bifurcation. The resets at −50 and −60 mV result in trajectories that pass very near the two nullclines, and are therefore very slow. The reset at −80 mV is outside the ghost of the attraction basin so that its trajectory passes further away from the saddle-node bifurcation. The insets show the enlarged areas enclosed by the gray rectangles. The arrow on the I-axis of the f-I plot indicates the current that was used to draw nullclines and trajectories in the phase plane. The conventions for line colors is the same as in Fig. 1

Fig. 3

Phase diagram and associated traces illustrating two types of tonic spiking: tonic spiking with sharp reset (a) and tonic spiking with broad reset (b). The only modifications needed to change the neuron model (a sharp resets) into the neuron model (b broad resets) is an increase of the spike triggered adaptation b and increase of the voltage reset, V _r. In both cases, some degree of adaptation is seen, yet both traces do not belong to the continuously adapting class because there is no substantial adaptation beyond the first two inter-spike intervals. The convention for symbols, line colors and scale bars was the same as in Fig. 1

Fig. 4

Phase plane representation of eight firing patterns. Firing patterns observed during a step current stimulation are: a tonic spiking, b adaptation, c initial burst, d regular bursting, e delayed accelerating, f delayed regular bursting, g transient spiking and h irregular spiking. The voltage traces are shown with a scale bar that corresponds to 100 and 20 mV. The graphs on the left show the traces in the phase planes as a trajectory (blue line) in the two state variables (V(t), w(t)). The w-nullcline (green) is a straight line, the V-nullcline before current stimulation is the curved dashed black line, and in the presence of stimulation, the curved solid line (black). The stable fixed point in (g) is indicated with a black, filled circle, and all the other symbols refer to the same convention as in Figs. 1 and 2. Comparing b with c illustrates that reset points jumping above the V-nullcline lead to initial bursting. Comparing c with d clarifies that regular bursting is obtained when the first broad reset generate a trajectory that passes below at least one of the previous reset points

Fig. 8

Comparison of the AdEx with three types of cortical neurons on step current injections. From left to right: experimental traces (red), AdEx model (blue), and overlay of the traces during onset and offset of the current step. From top to bottom: cNA(a), cAD (b), and RS (c). The left scale bar shows 20 mV and 300 ms, the scale bar for the overlays shows 20 mV and 20 ms. The current injections corresponds to 150 pA for cNA, 105 pA for cAD, and 130 pA for RS. Only one of the five repetitions is shown for clarity. Across multiple repetitions of the same stimulus, the time of the first spike or the first interspike intervals may jitter around what is seen on this figure. The f-I curves (d, e, f) of the fitted models show a steeper slope for the interneurons (d and e cNA and cAD, respectively) and a slow, type-I slope for the RS cells (f)

Fig. 5

Irregular firing is chaos in the AdEx. a Spikes times of an irregular spiking model are shown for three different amplitudes of the stimulating current step. At medium current amplitude (I = 150 pA) it spikes without periodicity, this current amplitude was used to make b and c. b The numerical integration of an irregular spiking model depends heavily on the initial conditions, such that ζ grows exponentially with the number of spikes simulated. The stars denote a modification of the initial condition in w only, the diamonds is a modification in V and the circles a modification in both w and V . c A linear fit shows a slope of 2.56 (full line). The interval map between each interspike interval and the preceding one appears as a thin, continous curve (n = 1240 spikes). The parameters for the irregular spiking model are given in Table 1

Fig. 6

Parameter space exploration of the four bifurcation parameters. Tonic spiking in red, adapting in yellow, initial bursting in green, regular bursting in cyan, irregular spiking in black and accelerating in blue. The four-dimensional parameter space was reduced to six relevant planes: a adaptive time constant (τ _w = 100 ms) and negative a (a = −5 nS), b refractory time constant (τ _w = 5 ms) and negative a (a = −5 nS), c adaptive time constant (τ _w = 100 ms) and low a (a = 0.001 nS), d refractory time constant (τ _w = 5 ms) and low a (a = 0.001 nS), e adaptive time constant (τ _w = 100 ms) and high a (a = 30 nS), f refractory time constant (τ _w = 5 ms) and high a (a = 30 nS). The firing pattern was classified for injection of current twice the rheobase, according to the criteria exposed in Sect. 3

Fig. 7

Phase diagram and parameter space for a piecewise linear V-nullcline in the limit of separation of timescales. a The firing pattern predicted from the analytical expression given in the text for the limit of separation of timescales is shown with the same color scheme as in Fig. 6. b. Trajectories for reset points where V _r ⩽ V _T go to high potential with little change in w unless they must contour the V-nullcline. c When V _r > V _T the trajectories go along the V-nullcline only when the reset point is situated above the dV/dt = 0. In both cases, the function M plateaus at a value w _c = F(VT) − X(V). The trajectories follow the V nullcline at a distance X(V), as illustrated in the inset of b (C = 100 pF, g _L = 10 nS, E _L = −70 mV, V _T = −50 mV, a = 0 nS, Δ = 3, τ _w = 2000 ms)