Mean-Field Reductions of Spiking Neural Networks

csm_website3b_6534a69d1c — Figure 2. Bifurcation diagram and time series. Bifurcation diagrams (a-d) depict the state space of the network model in terms of the mean external input to the two populations (excitatory and inhibitory, respectively). (a) Bifurcation diagram for the mean-field model without adaptation displaying an up and a down state, an oscillatory regime \(LC_{EI}\) (white solid contour) and a bi-stable region (green dotted contour. (b) Corresponding diagram for the ground-truth spiking model. (c) Bifurcation diagram for the mean-field model with adaptation, where the bi-stable region from (a) is replaced by a slow oscillatory region \(LC_{aE}\). (d) Corresponding diagram for the spiking network model. (e) and (f) display the firing rates time series (red excitatory population and blue inhibitory population) for the points A2 and B3 in the respective diagrams in (a-d).

While these reduced models of networks comprised of aEIF point neurons have proven very useful in the analysis of network dynamics, they are unable to distinguish basal-somatic from apical dendritic inputs and cannot account for an extracellular field because of their lack of spatial detail. Therefore, we employ a pyramidal neuron model that comprises two compartments to capture the above effects in a biophysically minimalistic way [Ladenbauer and Obermayer, 2019]. Using an analytical approach parameters are fitted to reproduce the response properties of a spatial model neuron and explore the stochastic spiking dynamics of single cells and large networks. Furthermore, a mean-field reduction is developed, based on the above Fokker-Planck approach, that allows for an accurate approximation of the firing rate dynamics of a network of coupled excitatory and inhibitory cells [Ladenbauer and Obermayer, 2019].

References

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[Augustin et al., 2017] Augustin, M., Ladenbauer, J., Baumann, F., & Obermayer, K. (2017). Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: comparison and implementation. PLoS Computational Biology, 13(6), e1005545.

[Brette and Gerstner, 2005] Brette, R. and Gerstner, W. (2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of Neurophysiology, 94(5):3637-3642.

[Cakan and Obermayer, 2019] Cakan, C., & Obermayer, K. (2019). State-dependent effects of electrical stimulation on populations of excitatory and inhibitory neurons. arXiv preprint arXiv:1906.00676.

[Ladenbauer et al., 2013] Ladenbauer, J., Augustin, M., and Obermayer, K. (2013). How adaptation currents change threshold, gain, and variability of neuronal spiking. Journal of Neurophysiology, 111(5):939-953.

[Ladenbauer and Obermayer, 2019] Ladenbauer, J., & Obermayer, K. (2019). Weak electric fields promote resonance in neuronal spiking activity: Analytical results from two-compartment cell and network models. PLoS Computational Biology, 15(4), e1006974.

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