## Abstract

The intrinsic electrophysiological properties of single neurons can be described by a broad spectrum of models, from realistic Hodgkin-Huxley-type models with numerous detailed mechanisms to the phenomenological models. The adaptive exponential integrate-and-fire (AdEx) model has emerged as a convenient middle-ground model. With a low computational cost but keeping biophysical interpretation of the parameters, it has been extensively used for simulations of large neural networks. However, because of its current-based adaptation, it can generate unrealistic behaviors. We show the limitations of the AdEx model, and to avoid them, we introduce the conductance-based adaptive exponential integrate-and-fire model (CAdEx). We give an analysis of the dynamics of the CAdEx model and show the variety of firing patterns it can produce. We propose the CAdEx model as a richer alternative to perform network simulations with simplified models reproducing neuronal intrinsic properties.

## 1 Introduction

Computational modeling of large-scale networks requires compromising between biophysical realism and computational cost. This requirement might be satisfied by many two-variable models (Morris & Lecar, 1981; Krinskii & Kokoz, 1973; Fitzhugh, 1961; Izhikevich, 2003; Brette & Gerstner, 2005). In particular, two-variable models that are largely used and studied are the Izhikevich model (Izhikevich, 2003) and the adaptive exponential integrate-and-fire (AdEx) model (Brette & Gerstner, 2005; Naud, Marcille, Clopath, & Gerstner, 2008; Touboul & Brette, 2008). The first variable of these models corresponds to the membrane voltage; the second, usually with slower kinetics, corresponds to neural adaptation and allows achieving more complex dynamics and firing patterns, which can be observed in neural recordings. In both models, the second variable has a form of an additional current flowing into the neuron, which may lead to unrealistic changes of membrane voltage, especially in the case of long and intense neuronal firings such as that observed during seizures (McCormick & Contreras, 2001). Here, we describe this limitation and propose a modification of the second variable dynamics to oppose it. The adaptation variable in our model has the form of a conductance, introducing the conductance-based adaptive experimental integrate-and-fire (CAdEx) model. Previous work has used conductance-based adaptation, but these models used were either more simplified (Treves, 1993) or much more detailed Hodgkin-Huxley-type models with complex channel gating dynamics (Connor & Stevens, 1971). A universal phenomenological model linking f-I curves with adaptation dynamics (Benda & Herz, 2003) has a useful general form, but it did not specify voltage dynamics and post-spike adaptation. The CAdEx model keeps the simplicity of the two-variables models, while extending the repertoire of possible subthreshold dynamics.

## 2 Conductance-Based Adaptive Exponential Model

A spike is initiated when $V$ approaches $VT$ and the exponential term escalates. After reaching a detection limit $VD$, the voltage is reset to the reset potential $VR$, and it remains at this value during a refractory period $\Delta tref$. After each spike, $gA$ is incremented by a quantal conductance $\delta gA$.

The sigmoidal subthreshold adaptation function is always positive, $g\xafA\u22650$, and it can be a monotonically increasing, $\Delta A>0$, or decreasing, $\Delta A<0$, function of membrane voltage. In the simulations, the value of the detection limit has been set to $VD=-40mV$ and the value of the refractory period to $\Delta tref=5ms$.

In appendix A4, we exemplify the parameterizations for I$M$ and I$h$ currents based on biophysical data (Brown & Adams, 1980; Banks et al., 1993).

The problem of unrealistic hyperpolarization in the AdEx model does not appear in the CAdEx model (see Figure 1). The corresponding problem of unrealistic hyperpolarization can appear also at the network level for AdEx neurons (see Figure 1b). Although both models can be tuned to exhibit very similar raster plots at the network level, the hyperpolarization problem disappears in networks of CAdEx neurons (see Figure 1). For a negative adaptation parameter $a$, corresponding to the accelerating firing pattern (Brette & Gerstner, 2005), in the AdEx model, an additional problem of infinite hyperpolarization can appear, as shown in appendix A6. Thus, the CAdEx model may be advantageous when the phenomena considered depend on the membrane potential.

All simulations of the models were done in Brian2 neural simulator and Python 3 programming language. The code, which allows running the CAdEx model simulations can be downloaded from https://github.com/neural-decoder/cadex.

## 3 Dynamical Analysis of the Model

### 3.1 Fixed Points and Bifurcations

The shape and the location of the $V$-nullcline depend on the input current $Is$ (see Figure 11 in appendix A1). The $V$-nullcline is divided by the vertical asymptote $V=EA$ into the left and right branches (see Figure 2a.)

The analysis of equation 3.2, which we can write in a simpler form defining a new function $S(V)$ as $Is=S(V)$, gives us the number of fixed points for a given input current (see Figure 2). The function S(V) tends to $-\u221e$ for $V\u2192\xb1\u221e$ and it can have:

A single maximum $Vmax$, which corresponds to the configuration with at most two possible intersections between nullclines. Let $ISN=S(Vmax)$. For a current input $Is<ISN$, there are therefore two fixed points $V-(Is)$ and $V+(Is)$; for $Is=ISN$, there is one fixed point; and for $Is>ISN$, there is no fixed point. $ISN$ is therefore the current above which the system always spikes spontaneously. The system can start to spike spontaneously for $Is<ISN$ if all fixed points lose stability before reaching $ISN$ in an Andronov-Hopf bifurcation (see below).

Two local maxima $Vmax1,2$, and one local minimum $Vmin$, which corresponds to maximally four possible fixed points. For input current $Is<S(Vmin)$, there are two fixed points, one stable and one unstable. For $S(Vmin)<Is<minSVmax1,2$, there are four fixed points, from which two can be stable. For $minSVmax1,2<Is<maxSVmax1,2$, there are two fixed points, and for $Is>maxSVmax1,2$, there is no fixed point and the neuron fires spontaneously.

To analyze the bifurcations of the system, we need to consider the behavior of the Jacobian matrix at the fixed points $i$ as a function of the input current, $Ji(Is)$ (see appendix A2).

The trajectories in the phase space of trace $trJi(Is)=tr(Ji(Is))$ and determinant $detJi(Is)=det(Li(Is))$ of the Jacobian give us the type of bifurcation (see Figures 3 and 4).

The trajectories of the equilibria and the corresponding dynamics in the CAdEx model are much different than they are in the AdEx model. In the AdEx model, at most only two fixed points are possible, and the trajectories of equilibria on the Poincaré diagram are linear (see appendix A.7).

### 3.2 Subthreshold Oscillations

The system can oscillate around equilibrium if the equilibrium is stable, that is, $trJ(Is)<0$ and $detJ(Is)>0$, and the eigenvalues of the Jacobian at the fixed point have an imaginary part, $trJ2-4detJ<0$. In that case, the frequency of oscillations is given by $\nu =14\pi 4detJ-trJ2$ (see Figure 5 and appendix A2).

The sustained subthreshold oscillations due to the emergence of a limit cycle can also be observed. Some examples in the case of multistability are given in the next section.

## 4 Multistability

In this section, we present a list of examples of multistability. Such behaviors can be observed due to two properties of the model:

*Subthreshold multistability*. Due to the nonlinear form of the subthreshold adaptation, the system may exhibit two or four fixed points. With four fixed points, various configurations are possible: with a positive or negative slope ($\Delta A$) of the adaptation nullcline. The stability of these fixed points, depending on the input current, is shown in Figure 4.*Superthreshold stability*. The model has a discontinuity that occurs after a spike and corresponds to the reset of the membrane voltage and the increment of adaptation conductance. It can lead to steady-state firing behavior, which can be treated as an attractor of the model. The reset discontinuity has, in many aspects, the same effect as a third variable. It allows chaotic behaviors (as described in the next section) normally impossible in a continuous two dimensional system.

Here, we give three examples of multistability: (1) with four fixed points and a negative slope of adaptation, (2) with four fixed points and a positive slope of adaptation, and (3) with two fixed points and a positive slope of adaptation:

With a negative and small enough value of $\Delta A$, the nullclines can have four crossings. In this case, the system has two stable fixed points with different values of the membrane potential. In this situation, three stable steady states are observed (see Figure 6a): two possible resting states of membrane potential and a self-sustained spiking state. Small perturbations permit a switch between these steady states, as shown in Figure 6b. By applying synaptic noise through conductance-based inhibitory and excitatory synapses (see appendix A3), transitions between these stable states can be observed (see Figure 6c).

With a positive and small enough value of $\Delta A$, the system can also have four fixed points. In this case, the emergence of a stable limit cycle is observed. It leads to three possible stable behaviors: resting state, subthreshold oscillations, and regular spiking, as shown in Figure 7a. As in the previous condition, transitions between these states can be observed under synaptic bombardment (see Figure 7b).

In a situation with two fixed points, the emergence of an unstable limit cycle is observed, leading to bistability near the threshold between spiking and nonspiking damped oscillations, as shown in Figure 8.

## 5 Firing Patterns

The CAdEx model is able to reproduce a large repertoire of intrinsic electrophysiological firing patterns. In this section, we detail the dynamics of firing patterns in CAdEx, reproducing those found in AdEx. In order to facilitate the use of this model to reproduce specific behaviors, we describe which parameters influence specific firing patterns. The initial values of the two variables in the simulations were $V(0)=-60mV$ for the membrane potential and $gA(0)=0nS$ for the adaptation, except when $\Delta A<0$ when $gA$ is initialized at $gA\u221e(-60mV)$. For the sets of parameters used in this section, the system does not have fixed points. Therefore, independent of the starting point location, the system converges to the same steady-state firing. However behaviors in the transient regime may change, which is crucial for spike frequency adaptation. The parameters for each pattern of Figure 9 are given in Table 1.

. | $Cm$ [pF] . | $EA$ [mV] . | $EL$ [mV] . | $Is$ [pA] . | $VA$ [mV] . | $\Delta A$ [mV] . |
---|---|---|---|---|---|---|

Adaptive spiking | 200 | −70 | −60 | 200 | −50 | 5 |

Tonic spiking | 200 | −70 | −70 | 192 | −45 | 5 |

Bursting | 200 | −60 | −58 | 150 | −45 | 1 |

Delayed bursting | 200 | −70 | −60 | 100 | −45 | 2 |

Accelerated spiking | 200 | −70 | −60 | 130 | −60 | −5 |

Chaotic-like spiking$1111$ | 200 | −70 | −58 | 90 | −40 | 5 |

$VR$ [mV]$1111111$ | $VT$ [mV] | $\delta gA$ [nS] | $g\xafA$ [nS] | $gL$ [nS] | $\tau A$ [ms] | |

Adaptive spiking$1111$ | −55 | −50 | 1 | 10 | 10 | 200 |

Tonic spiking | −56 | −50 | 0 | 2 | 10 | 40 |

Bursting | −46 | −50 | 1 | 10 | 10 | 200 |

Delayed bursting | −46 | −50 | 1 | 1 | 12 | 100 |

Accelerated spiking | −58 | −48 | 0 | 6 | 10 | 300 |

Chaotic-like spiking | −47 | −50 | 1 | 10 | 10 | 25 |

. | $Cm$ [pF] . | $EA$ [mV] . | $EL$ [mV] . | $Is$ [pA] . | $VA$ [mV] . | $\Delta A$ [mV] . |
---|---|---|---|---|---|---|

Adaptive spiking | 200 | −70 | −60 | 200 | −50 | 5 |

Tonic spiking | 200 | −70 | −70 | 192 | −45 | 5 |

Bursting | 200 | −60 | −58 | 150 | −45 | 1 |

Delayed bursting | 200 | −70 | −60 | 100 | −45 | 2 |

Accelerated spiking | 200 | −70 | −60 | 130 | −60 | −5 |

Chaotic-like spiking$1111$ | 200 | −70 | −58 | 90 | −40 | 5 |

$VR$ [mV]$1111111$ | $VT$ [mV] | $\delta gA$ [nS] | $g\xafA$ [nS] | $gL$ [nS] | $\tau A$ [ms] | |

Adaptive spiking$1111$ | −55 | −50 | 1 | 10 | 10 | 200 |

Tonic spiking | −56 | −50 | 0 | 2 | 10 | 40 |

Bursting | −46 | −50 | 1 | 10 | 10 | 200 |

Delayed bursting | −46 | −50 | 1 | 1 | 12 | 100 |

Accelerated spiking | −58 | −48 | 0 | 6 | 10 | 300 |

Chaotic-like spiking | −47 | −50 | 1 | 10 | 10 | 25 |

*Adaptive spiking.* The adaptation can cause a decrease of spiking frequency (see Figure 9a). Correspondingly, the value of $gA$ increases until it reaches a steady state. The spike frequency adaptation is mainly affected by the spike triggering parameters ($VT$ and $\Delta T$); by the membrane properties, $C$ and $gL$; as well as by the adaptation parameters, $g\xafA$ and strongly by $\delta gA$.

*Tonic spiking.* In the absence of adaptation, that is, $g\xafA=0$, the model exhibits tonic spiking. However, tonic spiking can also be observed with nonzero adaptation, as in Figure 9b, if the steady-state firing can be reached after the first spike. The steady-state spiking frequency is influenced by the membrane capacitance, $C$, the leak conductance, $gL$, and the slope of spike initiation $\Delta T$, and also by the adaptation time constant $\tau A$. The after-hyperpolarization following spiking depends on both: reset value $VR$ and adaptation time constant $\tau A$.

*Bursting.* To obtain bursting behaviors (see Figure 9c), the reset value, $VR$, has to be higher than the voltage of the minimum of the $V$-nullcline. The system spikes until it crosses the $V$-nullcline, ending the burst. Then, above the $V$-nullcline, the system is driven to lower values of adaptation and membrane potential. After a second crossing of the $V$-nullcline, the system spikes again, starting a new bursting cycle. Bursts can be characterized by intra- and interburst activities. The interburst activity is affected by adaptation; $g\xafA$ and $\tau A$ strongly determine the interburst time interval and after-burst hyperpolarization.

*Delayed spiking.* All firing patterns can occur with a time delay. Figure 9d gives an example for bursting and Figure 9b an example for low-frequency spiking. The distance between $V$ and $gA$ nullclines is determined by $Is$. By changing this distance and the time constant $\tau A$, it is possible to obtain a region of slow flow (small $dVdt$) and consequently to increase the delay and the interevent interval. The reset $VR$ and $\delta gA$ also affects the time required for the system to go around the $V$ nullcline and then to the next event.

*Accelerated spiking.* A negative slope of $gA$ ($\Delta A<0$) associated with $\delta gA=0$, leads to firing rate acceleration, as shown in Figure 9e.

*Chaotic-like spiking.* Because the CAdEx model is not continuous due to the after-spike reset of voltage and incrementation of adaptation, chaotic-like spiking behavior can be observed, as shown in Figure 9f. These phenomena occur when the reset is close to the right branch of the $V$ nullcline. To verify that this irregularity is not due to numerical error, we used various integration methods and various time steps in the Brian2 simulator (Stimberg, Brette, & Goodman, 2019).

## 6 Adaptation and Irregularity

Both subthreshold $g\xafA$ and postspike $\delta gA$ adaptation parameters affect the adaptation index (see Figure 10a). This allows the model to reproduce the wide range of $A$ index values observed in neurons (Allen Brain Institute, 2015).

The irregular spiking (like chaotic-like spiking and bursting) is especially pronounced in the transition zone between slow and fast regular spiking regions (see Figure 10b). On the phase diagram, slow, regular spiking corresponds to a postspike reset occurring on the left side of the right branch of the V-nullcline and, consequently, leading to longer interspike intervals, while fast tonic spiking corresponds to a reset occurring on the right side, leading to fast subsequent spiking. In the transition zone, alternations between resets on the left and right sides of the V-nullcline can lead to highly irregular spiking (see the chaotic-like spiking pattern Figure 9f) and bursting (see Figures 9c and 9d).

This irregularity of spiking originates from the postspike reset mechanism, which also occurs in the AdEx model (Touboul & Brette, 2008). The correspondence between parameters of both models is intuitive: $\delta gA$ corresponds to $b$ in the AdEx model, while the reset potential $VR$ has the same meaning in both models.

## 7 Discussion

In this article, we have proposed a new integrate-and-fire model with two variables, which can produce a large repertoire of electrophysiological patterns while still allowing for clear mathematical insights and large-scale simulations. This CAdEx model is completely specified with 12 biophysical parameters, and reproduces qualitatively similar pattern as the AdEx model (Naud et al., 2008), because the $gA$ nullcline may be considered as locally linear and approximates that of the AdEx model. While the dynamics of the CAdEx model is comparable to the AdEx model for moderate input and firing, the CAdEx model does not suffer from an unnaturally strong hyperpolarization after prolonged periods of strong firing. This can be very advantageous for modeling of highly synchronized rhythms and firings, like slow-wave oscillations or epileptic seizures. The different behaviors of the averaged membrane voltages between the models were demonstrated here at the network level. Moreover, the sigmoidal subthreshold adaptation function allows one to model the dynamics of voltage-dependent ion channels in more detail while retaining the overall computational simplicity. The sigmoidal form of the adaptation function enriches the dynamics, allowing a wider repertoire of multistabilities. An important difference between CAdEx and AdEx models is that the CAdEx model includes a new bifurcation structure with an Andronov-Hopf bifurcation in the four-fixed-point configuration (see Figure 4, upper right). In this regime, there are two simultaneously stable fixed points—one can be in an “integrator” mode (and so going through saddle-node bifurcation when losing stability) with the other one in a “resonator” mode—and the strongly resonating fixed point can then undergo an Andronov-Hopf bifurcation (Izhikevich, 2007). Thus, the CAdEx model can display a bistability between integrator and resonator modes for the same set of parameters. It would be interesting to see if such a predicted bistability can be found in other models or experimentally.

The CAdEx model also solves a problem of divergence of the AdEx model to minus infinity for a negative slope of subthreshold adaptation. This divergence problem can be a source of instabilities in network simulations. It is automatically fixed by the CAdEx model because the voltage is bound to the reversal value of the adaptation current.

Our model also has some limitations. First, the adaptation has the form of a noninactivating current (such as I$M$ potassium current), which limits the description of a class of inactivating ionic channels. It also includes only one type of subthreshold adaptation. In comparison to the AdEx model, the computational cost of our model may be slightly higher due to the form of the adaptation variable and, more specifically, the introduction of an exponential function. Also, as more parameters are introduced, more thorough dynamical studies and explorations of the parameter space are needed.

## Appendix

### A.1 Shape of V-Nullclines While Varying Input Current

The V-nullcline changes its shape when the input current $Is$ changes. When the input current is strong and hyperpolarizing, the V-nullcline is inverted near an asymptote $V=EA$ (see Figure 11).

### A.2 Bifurcation Analysis

#### A.2.1 Minima and Maxima of the $S(V)$ Function

A single global maximum of $S(V)$ function, $S(Vmax)$, for two possible intersections between the $V$ and $gA$ nullclines.

Two maxima and one minimum of $S(V)$ function, for four possible intersections between the $V$ and $gA$ nullclines.

#### A.2.2 Local Linearized Dynamics around Equilibria

### A.3 Conductance-Based Synapses

### A.4 I$M$ and I$h$ Currents

The currents I$M$ and I$h$ are proposed as an example of currents that can be simulated using the CAdEx model. For the hyperpolarizing current I$h$, the parameters have been identified in Banks et al. (1993), and for I$M$, the parameters have been identified in Brown and Adams (1980). The values of parameters for these currents are listed in Table 2.

. | $Cm$ [pF] . | $EA$ [mV] . | $EL$ [mV] . | $Is$ [pA] . | $VA$ [mV] . | $\Delta A$ [mV] . |
---|---|---|---|---|---|---|

Neuron with $IM$ | 200 | −90 | −60 | 350 | −35 | 4 |

Neuron with $Ih$$1111$ | 200 | −43 | −60 | 100 | −75.7 | −5.7 |

$VR$ [mV]$1111111$ | $VT$ [mV] | $\delta gA$ [nS] | $g\xafA$ [nS] | $gL$ [nS] | $\tau A$ [ms] | |

Neuron with $IM$$1111$ | −58 | −45 | 1 | 16 | 10 | 550 |

Neuron with $Ih$ | −70 | −50 | 1.5 | 43 | 12 | 800 |

. | $Cm$ [pF] . | $EA$ [mV] . | $EL$ [mV] . | $Is$ [pA] . | $VA$ [mV] . | $\Delta A$ [mV] . |
---|---|---|---|---|---|---|

Neuron with $IM$ | 200 | −90 | −60 | 350 | −35 | 4 |

Neuron with $Ih$$1111$ | 200 | −43 | −60 | 100 | −75.7 | −5.7 |

$VR$ [mV]$1111111$ | $VT$ [mV] | $\delta gA$ [nS] | $g\xafA$ [nS] | $gL$ [nS] | $\tau A$ [ms] | |

Neuron with $IM$$1111$ | −58 | −45 | 1 | 16 | 10 | 550 |

Neuron with $Ih$ | −70 | −50 | 1.5 | 43 | 12 | 800 |

### A.5 Network Simulations

The parameters of the AdEx and CAdEx networks for which results are presented in Figure 1 are shown in Tables 3, 4, and 5. The input to both networks was given by current noise injected to all neurons and had the form of an Ornstein-Uhlenbeck process, with the same parameters: a variance, $\sigma =2.4pA$, and a time constant, $\tau =100ms$. The neurons were connected by conductance-based synapses (see appendix A3).

. | . | AdEx . | |
---|---|---|---|

. | . | Excitatory Cells . | Inhibitory Cells . |

Population | N | 800 | 200 |

Capacitance | $C$ | 150 pF | |

Leak conductance | $gL$ | 10 nS | |

Leak reversal potential | $EL$ | −63 mV | −65 mV |

Spike threshold | $VT$ | $-$50 mV | |

Spike initiation slope | $\Delta T$ | 2 mV | 0.5 mV |

Adaptation time constant | $\tau w$ | 500 mV | n.a. |

Post-spike adaptation increment | $b$ | 107 pA | n.a. |

Subthreshold adaptation | $a$ | 0 nS | n.a |

Quantal synaptic conductance | $QE/I$ | 1.2 nS | 5 nS |

Synaptic reversal potential | $EE/I$ | 0 mV | −75 mV |

. | . | AdEx . | |
---|---|---|---|

. | . | Excitatory Cells . | Inhibitory Cells . |

Population | N | 800 | 200 |

Capacitance | $C$ | 150 pF | |

Leak conductance | $gL$ | 10 nS | |

Leak reversal potential | $EL$ | −63 mV | −65 mV |

Spike threshold | $VT$ | $-$50 mV | |

Spike initiation slope | $\Delta T$ | 2 mV | 0.5 mV |

Adaptation time constant | $\tau w$ | 500 mV | n.a. |

Post-spike adaptation increment | $b$ | 107 pA | n.a. |

Subthreshold adaptation | $a$ | 0 nS | n.a |

Quantal synaptic conductance | $QE/I$ | 1.2 nS | 5 nS |

Synaptic reversal potential | $EE/I$ | 0 mV | −75 mV |

. | . | CAdEx . | |
---|---|---|---|

. | . | Excitatory Cells . | Inhibitory Cells . |

Population | N | 800 | 200 |

Capacitance | $C$ | 150 pF | |

Leak conductance | $gL$ | 10 nS | |

Leak reversal potential | $EL$ | −63 mV | −65 mV |

Spike threshold | $VT$ | $-$50 mV | |

Spike initiation slope | $\Delta T$ | 2 mV | 0.5 mV |

Adaptation time constant | $\tau A$ | 500 mV | n.a. |

Postspike adaptation increment | $\delta gA$ | 5 nS | n.a. |

Maximal subthreshold adaptation | $g\xafA$ | 0 nS | n.a |

Adaptation reversal potential | $EA$ | −70 mV | n.a |

Quantal synaptic conductance | $QE/I$ | 1.2 nS | 5 nS |

Synaptic reversal potential | $EE/I$ | 0 mV | −75 mV |

. | . | CAdEx . | |
---|---|---|---|

. | . | Excitatory Cells . | Inhibitory Cells . |

Population | N | 800 | 200 |

Capacitance | $C$ | 150 pF | |

Leak conductance | $gL$ | 10 nS | |

Leak reversal potential | $EL$ | −63 mV | −65 mV |

Spike threshold | $VT$ | $-$50 mV | |

Spike initiation slope | $\Delta T$ | 2 mV | 0.5 mV |

Adaptation time constant | $\tau A$ | 500 mV | n.a. |

Postspike adaptation increment | $\delta gA$ | 5 nS | n.a. |

Maximal subthreshold adaptation | $g\xafA$ | 0 nS | n.a |

Adaptation reversal potential | $EA$ | −70 mV | n.a |

Quantal synaptic conductance | $QE/I$ | 1.2 nS | 5 nS |

Synaptic reversal potential | $EE/I$ | 0 mV | −75 mV |

. | Excitatory Cells . | Inhibitory Cells . |
---|---|---|

Excitatory cells | 12% | 10% |

Inhibitory cells | 10% | 12% |

. | Excitatory Cells . | Inhibitory Cells . |
---|---|---|

Excitatory cells | 12% | 10% |

Inhibitory cells | 10% | 12% |

### A.6 Unbounded Hyperpolarization of the AdEx Model

### A.7 Mathematical Overview of the AdEx Model

## Acknowledgments

The work was supported by CNRS, the European Community (Human Brain Project, H2020-785907), and the ICODE excellence network.

## References

## Author notes

T.G. and D.D. contributed equally.