Fig 1.
Variables and interactions that must be present in the models to capture all relevant behaviors, the main molecules associated to each of the variables are also displayed.
The type of interaction is marked on the arrow. For instance, w modulates (M) V as it influences the intrinsic dynamics of V but does not usually cause it directly. On the other hand, as changes in the membrane potential are the main cause of variations in w, V is said to drive (D) w. Eventually, all mechanisms consume (C) energy.
Fig 2.
Phase space of the eLIF model in bistable parameter regime.
V-nullcline is given by the blue line, ϵ-nullcline by the red curve. Fixed points (FPs) are shown by the circles (filled for stable and empty for unstable) and the cross marks the inflection point of the ϵ-nullcline. Dashed lines represent the shifts in the V-nullcline which lead to the disappearance of the unstable fixed point and of one of the stable fixed points (saddle-node bifurcation via the external current Ie). The super-threshold region, where spikes are elicited upon entrance, is marked by the light grey shading; the energy-limiting region (ϵ < ϵc) is marked by the grey shading and overlaps with the super-threshold region in the dark grey area, where energy limitations prevent spiking though the neuron is above threshold.
Fig 3.
Dynamics of the eLIF model as timeseries (A) and in phase-space (B) in the bistable regime.
A. The behavior of the model is shown in response to four different inputs, shown in grey on the V subplot: a low depolarizing current (a: 10 pA), a stronger depolarizing current (b: 30 pA), a large depolarization (c: 80 pA), and a hyperpolarizing current (d: -60 pA). For visualization purposes, action potentials are made visible by setting the voltage to -50 mV at spike time. B. Corresponding behavior in phase-space is shown, each subplot corresponding to one of the four domains separated by the grey dashed lines on panel A. The black curves mark the resting nullclines and the light grey line marks the input-driven V-nullcline; resting fixed points (FPs) are marked by the large black circles while input-driven FPs are show by the small grey circles and spike emissions are marked by empty left triangles while reset positions are marked by blue dots. The neuron displays the following behaviors: (a) subthreshold dynamics, where the neuron temporarily leaves the high-energy FP, associated to the down-state, then goes back towards it, (b) transition from the initial high-energy FP to the low-energy FP (up-state) through a spiking period, (c) transition from the up-state to a depolarization block via a spiking period before returning towards the up-state, (d) transition from up- to down-state. See S2 Table for detailed parameters.
Fig 4.
One possible pathway for the transition between healthy and diseased state in the eLIF model.
In the model, progressive decrease in the “energetic health” factor α, from 1 to 0.3, leads to a succession of changes in both the number of fixed points (FPs) and in their properties. The middle panel shows the evolution of the FPs’ energy levels—filled circles for stable FPS, empty for unstable FPs—with the grey line marking ϵc. Four stages of the disease progression are also illustrated in phase-space: (a) healthy neuron with a single FP. (b) bifurcation to a 3 FPs state without major changes in the dynamical properties (susceptible but potentially “asymptomatic” cell). (c) bifurcation to a single low-energy FP associated to an extremely excitable state (diseased cell). (d) further decrease of the energetic health brings the FP below the energy threshold ϵc, leading the neuron to become unresponsive. In stages (a) and (b), the neuron lies in its resting state in the absence of input; however, at stages (c) and (d), the two insets on the upper panel show the membrane dynamics of the neuron for a hypothetical “accelerated evolution” of the disease, where the neuron respectively enters (35-second simulation) and leaves (45-second simulation) the “hyperactive” region where usually subthreshold inputs (here modeled by a Poisson noise) are sufficient to trigger uncontrolled spiking. See S2 Table for detailed parameters.
Fig 5.
Decrease in health can be partly compensated by homeostatic mechanisms and be invisible from the statistical properties of background activity, as shown by the behavior on an excitatory and inhibitory population with N = 1000 neurons in the asynchronous irregular (AI) state.
A. For such a network, changes in the neuronal health, modeled by a decrease in the α parameter, do not appear in the background activity of the raster (non-grayed areas), where the activity of both excitatory (red circles) and inhibitory (gray triangles) neurons remain very similar. To see the actual consequences of the decrease in health, one must look at the response of the network to an additional input, which is shown in the grayed areas on panel A. In response to a threefold increase in the rate of Poisson input between 1400 and 1650 ms, the activity of 100 excitatory neurons (marked by the orange area) progressively switches from continuous tonic firing (top) to well-separated bursts (bottom). B. More quantitative analyses also confirm that the background activity remains close to Poissonian, with coefficients of variation (CVs) around 0.7–0.8, and asynchronous, with an average cross-correlation (CC) smaller than . C. The distributions of firing rates over 5 seconds remain almost identical and centered around 2 Hz; dotted and dashed lines respectively denote the quartiles and medians.
Fig 6.
Typical dynamics of the mAdExp model with different parameter settings in response to current steps given by the scale bars—500 ms for all entries—In yellow to mark lower excitation, red to mark higher excitation, blue bar and asterisk on IR to mark inhibitory current.
The behaviors include regular spiking (RS), adaptive spiking (AS), initial burst (IB), regular bursting (RB), transient spiking (TS), delayed bursting (DB), and delayed accelerating (DA). Similar responses to the lower (yellow) currents can be achieved by the original AdExp model. However, each of these dynamics now comes with an “energy-depleted” state for high input current (red), associated to a depolarization block (responses associated to red bars), that cannot be captured by AdExp model. In addition to these standard behaviors, dynamical repertoire of the mAdExp neuron also includes a different mechanism for post-inhibitory rebound spiking (IR), and can display post-excitatory rebound (ER) or intermittent spiking dynamics (IS). See S3 Table for detailed parameters.
Fig 7.
Voltage traces for two cell types (566978098 and 570896413 in Allen Brain Atlas) and associated fits with mAdExp and AdExp neuron models.
Fourth row represents the input current. Additional or missed spikes are marked in parentheses on the left of the associated spike train. Activities in the rectangles are expanded in the lower panels. A. Cell presenting little to no sag upon hyperpolarization and adaptive spiking behavior (A.1); expanded activity (A.2) enables to see the discrepancies between the AdExp model (green) and the data (thin black line) while mAdExp (blue) matches the dynamics much more precisely. B. Cell presenting significant sag upon hyperpolarization and almost immediate depolarization block upon depolarizing input (B.1). Both AdExp and mAdExp match the rebound dynamics; however, AdExp cannot reproduce the depolarization block as shown in the expanded dynamics (B.2). See 5 for detailed parameters.