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Alain Destexhe CNRS, Gif sur Yvette, April 2021. We overview here the investigations of networks using the Adaptive Exponential (AdEx) model of neurons in our research team. For a detailed description of the mean-field models of AdEx systems that we developed, see the section Mean-field models of neuronal populations. Brain statesOne of the goals of our network models is to reproduce the neuronal activity as seen in the functioning brain in vivo, so we have conceived these models so that they can be fit to experimental data. Thus, we start by showing experimental recordings and next we show different network models of the activity states that were recorded experimentally.The figure below shows recordings of human neurons with micro-electrode arrays (left). The micro-electrodes record a series of neurons which can be individualized (using classic spike-sorting methods). The neurons could also be discriminated between “regular-spiking” (RS) and “fast-spiking” (FS), which were shown to correspond essentially to excitatory and inhibitory neurons [1]. The figure shows recordings during two different brain states in human, slow-wave sleep (middle) and wakefulness (right). One can see that the dynamics in wakefuless is sustained, and neurons fire asynchronously and very irregularly. These dynamics are called “asynchronous-irregular” (AI) states, and is seen in all mammals when the brain is in active wakefulness. During slow-wave sleep, the dynamics is characterized by the alternance between periods of sustained firing, called “Up states” and periods of suppressed (or highly reduced) firing, called “Down states”. This Up/Down state dynamics are typical of the brain activity in all mammals during slow-wave sleep. In general, one can see that the inhibitory (FS) cells (in red) fire at higher rates compared to RS cells. This is also a feature generally seen in all mammals. 
Integrate-and-fire modelTo build networks, the simplest model is the integrate-and-fire (IF) model, where the neuron is described by a passive membrane with a threshold. When the threshold is reached, the neuron fires and its membrane potential is set to a reset value. Networks of IF neurons can be formed, and connected randomly with excitatory and inhibitory synapses. Such models are known to generate asynchronous-irregular (AI) states, as shown in the figure below [3].
AdEx modelNetworks of IF neurons can generate AI states, but cannot generate Up/Down states. To palliate to this caveat, we need to take into account slow variables, such as spike-frequency adaptation. This is why we have considered AdEx models, which is a neuron model slightly more complex than the IF neuron. The AdEx model is a two-dimensional IF model that includes an additional variable which can model slow adaptation (or bursting for some parameters). In particular, the AdEx model can simulate the RS and FS cells, as shown here. When connected into networks (with sparse random connectivity), they can exhibit AI states, where the FS cells fire at higher rates compared to RS cells, exactly like experimental data [4].
A mean-field model was derived for AdEx networks with RS and FS cells [4]. In this mean-field, the variables are the population firing rates of RS and FS cells. The figure below shows that the mean-field model captures well the time course of the response of the network to an external input. 
One of the interesting feature of AdEx networks is that they can also generate Up/Down state dynamics. The figure below shows the behavior of an AdEx network in two different states, the AI state (upper graphs) and slow oscillations with Up and Down states (lower graphs). Panel B shows the mean rate of RS and FS cells, and panel C shows the mean-field for comparison. One can see that the AdEx mean-field model can also generate Up/Down states. (see details in refs. [5,6]) 
The transition between those two states is illustrated in the figure below. The experiments (top graphs) show that the Up/Down state dynamics (as seen here from the EEG and the simultaneous intracellular recording of a cortical neuron) can switch to AI state. The switch was induced here by stimulating the pedunculo-pontine (PPT) nucleus which releases acetylcholine in the thalamocortical system. The AdEx model (bottom graph) can display a similar transition, if adaptation is reduced [5]. 
Computing with AI statesThe AdEx network models were used to investigate possible computational properties of AI states [7]. The figure below considers an AdEx network submitted to an external input (A). When one compares different states in the same AdEx network (B; AI state, synchronized oscillation or SR state, and quiescent state), the highest responsiveness is obtained in the AI state (C-D). This can be explained by the boosting effect of the “internal noise” present in the AI state.
This enhanced responsiveness of the AI state can be exploited by considering several network layers, each described by an AdEx network (see figure below, panel A). When the networks are set into an AI state, a given input propagates across the layers (B, blue), whereas if the networks are quiescent, the information vanishes (B, red; see ref. [7] for details). 
ConclusionThese studies have shown that the AdEx model permits to define two cell types, RS and FS cells, and the adaptation present in the AdEx model also allows to generate Up/Down state dynamics, as well as AI states. We found that AI states are the most responsive and can produce a response to weak stimuli, and it does so better than any other oscillatory or quiescent state [7]. When placed into several layers, the AdEx networks in AI state can propagate information, while the information does not propagate if the networks are silent. This underlies the importance of knowing what is the “ongoing activity” state of the system to understand how information propagates in real neural networks. We suggest that this mechanism of AI-driven information processing is at play in the awake brain, which typically displays AI states. We are presently investigating how these properties are expressed at the level of large networks, up to the whole brain [6].References |