Biophysically grounded mean-field models of neural populations under electrical stimulation

Abstract

Electrical stimulation of large neural networks is a key tool for understanding neural population dynamics and ultimately for developing clinical treatments. However, computational models of cortical networks of spiking neurons are inherently hard to simulate and analyze. We present an efficient mean-field model of excitatory and inhibitory adaptive exponential integrate-andfire (AdEx) neurons of which we analyse the dynamical properties and validate the results using detailed network simulations. Bifurcation diagrams reflecting the network's state reveal asynchronous up and down-states, bistable regimes, and oscillatory regions corresponding to fast excitation-inhibition and slow excitation-adaptation feedback loops. In this dynamical landscape, electrical stimuli can cause state transitions, such as turning on and off oscillations. Oscillatory input can frequency-entrain and phase-lock endogenous oscillations. The effects of time-varying external stimulation are well-predicted by the mean-field model, further underpinning the utility of low-dimensional neural mass models. ResultsThe cortical mass model Here we study a cortical mass model which consists of two populations of excitatory adaptive (E) and inhibitory (I) exponential integrate-and-fire (AdEx) neurons (Fig. 1). Both populations are delay-coupled and the excitatory population has a somatic adaptation feedback mechanism. The low-dimensional mean-field model (Fig. 1 a) is derived from the large network of spiking AdEx neurons (Fig. 1 b).For the construction of the mean-field model, a set of conditions need to be fulfilled: We assume the number of neurons to be very large, all neurons within a population to have equal properties, and the connectivity between neurons to be sparse and random. Additional assumptions about the mathematical nature and a detailed derivation of the mean-field model is presented in the Methods. Bifurcation diagrams: a map of the dynamical landscapeThe E-I motif shown in Fig. 1 can occupy various network states, depending on the baseline inputs to both populations. By gradually changing the inputs, we map out the state space of the system, depicted in the bifurcation diagrams in Fig. 2. Small changes of the parameters of a nonlinear system can cause sudden and dramatic changes of its overall behavior, called bifurcations. Bifurcations separate the state space into distinct regions of network states between which the system can transition from one to the other. In our case, the dynamical state of the E-I system depends on external inputs to both subpopulations, which are directly affected by external electrical stimulation and other driving sources, e.g. inputs from other neural populations such as other brain regions.Comparing the bifurcation diagrams of the mean-field model (Figs. 2 a, c) to the ground truth spiking AdEx network (Figs. 2 b, d) demonstrates the similarity between both dynamical landscapes. Transitions between states take place at comparable baseline input values and in a well-preserved order. 2/25 11/25

Neurostimulation stabilizes spiking neural networks by disrupting seizure-like oscillatory transitions

“…In contradistinction to these cell-type specific experiments that can be performed in vitro and in vivo in transgenic animals, the clinical application of neuromodulation is currently limited to the non-specific delivery of electrical current to the brain. While the real-world clinical benefits of neuromodulation have been demonstrated both for epilepsy 3 – 7 and for movement disorders using deep brain stimulation (DBS) 16 , 19 , 20 , our understanding of how these effects are realized in neuronal circuits remains speculative at best 17 , 18 . In this context it has been opined that “neurostimulation for epilepsy often has moved retrograde, from pilot clinical studies back to the laboratory for validation and modification of stimulation methods” 21 .…”

neurolib: a simulation framework for whole-brain neural mass modeling

“…The bifurcation diagram of the ALN model in Fig. 3 g has a more complex structure and its validity has been verified using large spiking network simulations before 24 . Here, we can see that the down-state and the up-state are separated by a limit cycle as well.…”

“…3 i, where the activity shows noise-induced transitions between the up- and down-state. When the adaptation mechanism of the underlying AdEx model is enabled (not shown), the bistable region transforms into a new, slowly oscillating limit cycle 24,34 .…”

“…In order for the model to produce slow oscillations that fit the data, we included the spike-frequency adaptation strength parameter b and the adaptation time scale τ A in our optimization, culminating in a total of six free parameters. In a single node, adaptation-induced oscillations can have a frequency between roughly 0.5 and 5 Hz 24 . Figure 5 b shows the location of all good fits in the bifurcation diagram (FC and FCD thresholds as above, EEG power spectrum correlation > 0.7).…”

The Effect of Alterations of Schizophrenia-Associated Genes on Gamma Band Oscillations

“…Furthermore, a search for novel therapeutic targets can be conducted through a more exploratory analysis of the effect of parameters on the model behaviour (also see [86]). Additionally, such an approach is not limited to pharmacological interventions since transcranial electric or magnetic stimulation can easily be incorporated (as demonstrated in other modeling studies such as [10,85,82]); this is, of course, not restricted to schizophrenia but can be applied to psychiatric disorders in general.…”