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Mean-Field Reductions of Spiking Neural Networks

Let us consider a network consisting of \( N=N_E + N_I\) neurons, where \(N_E\) and \(N_I\) are the numbers of excitatory and inhibitory neurons, respectively (a schematic is depicted in Figure 1; taken from [Cakan and Obermayer, 2019]). Each single neuron is modelled using the adaptive exponential integrate and fire (aEIF) model [Brette and Gerstner, 2005], where the dynamics of the neuron is described by

$$ C \frac{dV}{dt}=I_{ion}-w+I_{syn} \tag{1} $$

$$ \tau_w \frac{dV}{t}=a(V-E_L)-w \tag{2} $$

with a reset condition if \(V>C_{cut}\) then \(V:=V_r \) and \(w:=w+b\). 

Here, \(V\) represents the membrane potential of the neuron, \(w\) the adaptation current, \(C\) the membrane capacitance, \(I_{ion}\) the sum of ionic currents, \(I_{syn}\) the synaptic current, \(\tau_w\) the adaptation time constant, \(a\) the subthreshold adaptation conductance, \(E_L\) the leak conductance, \(V_{cut}\) the spike ‘cuttoff’ value, \(V_r\) the reset potential and \(b\) the adaptation increment that mediates spike-triggered adaptation. This neuron model has been shown to reproduce a wide variety of properties of cortical neurons [Touboul and Brette, 2008, Naud et al., 2008] and its adaptation parameters have been linked to potassium currents in a more biophysically detailed neuron model [Ladenbauer et al., 2013]. 

Figure 1. Schematic of the cortical motif. Coupled populations of excitatory (red) and inhibitory (blue) neurons. (a) Mean-field neural mass model, where each node represents a population. (b) Corresponding spiking network model (based on the aEIFmodel). Both populations receive independent input currents with a mean of \(\mu_{ext}\) and a standard deviation of \(\sigma_{ext}\).
Lupe

A single cell \(i\) of population \(\alpha\) receives a total synaptic current

$$ I_{syn,i}^{\alpha} (V_i^{\alpha},t) := \sum_j I_{ij}^{\alpha,ext} + \sum_j I_{ij}^{\alpha,E} + \sum_j I_{ij}^{\alpha,I}  \tag{3}$$

which is the sum over the synaptic currents \( I_{ij}^{\alpha,ext} \) from \(K_{ext}\) external excitatory neurons, \(I_{ij}^{\alpha,E}\) from \(K_{E}\)  excitatory neurons in the network and \(I_{ij}^{\alpha,I}\) from \(K_{I}\) inhibitory neurons in the network. Here, \(j\) is the index of the respective presynaptic neuron. Synaptic currents are modelled according to

$$  I_{ij}^{\alpha,ext} (V_i^{\alpha},t) := C J_{ij}^{\alpha,ext} \sum_k \delta (t-t_j^k)(E_E -V_i^{\alpha}) \tag{4}$$

for the population of external excitatory neurons and

$$ I_{ij}^{\alpha,\beta} (V_i^{\alpha},t) := C J_{ij}^{\alpha,\beta} \sum_k \delta (t-t_j^k-d_{ij}^{\alpha,\beta})(E_{\beta} -V_i^{\alpha}) \tag{5}$$

where \(\alpha \) denotes the postsynaptic population and \( \beta \)denotes the presynaptic population. Furthermore, \(J_{ij}^{\alpha,\gamma}\), with \(\gamma \in \{ext,E,I\}\), describe the synaptic efficacies which are drawn from Gaussian distributions with mean \(J_{\alpha,\gamma}\) and standard deviation \(\Delta J_{\alpha,\gamma}\). The time of the \(k\)-th spike of neuron \(j\) is denoted as \(t_j^k\). The excitatory and inhibitory reversal potentials are denoted as \(E_E\) and \(E_I\), respectively. Synaptic delays \( d_{ij}^{\alpha,\beta}\) are drawn from a bi-exponential probability density

 $$ p_d^{\alpha,\beta} (d) := \frac{1}{\tau_d-\tau_r} (e^{\frac{-d-d_0}{\tau_d}}-e^{\frac{-d-d_0}{\tau_r}}) \tag{6}$$

where \(d\) is a positive delay, \(d_0\) is the minimal delay and \(\tau_r\), \(\tau_d\) are the rise and decay time constants, respectively. Each neuron in the external population generates spike times according to a Poisson process with rate \(r_{ext}^{\alpha}(t)\).

This network setup allows to readily apply the mean-field approach developed in [Augustin et al., 2013]. Here the total synaptic current \(I_{syn,i}^{\alpha}\) will be approximated as a sum of its mean and a time-varying Gaussian part

$$ I_{syn,i}^{\alpha} \approx \mu_{\alpha,i}(V_i^{\alpha},t)+\sigma_{\alpha,i}(V_i^{\alpha},t) \eta_i(t) \tag{7}$$

where \(\mu_{\alpha,i}\) and \(\sigma_{\alpha,i}\) are the infinitesimal mean and standard deviation of  \(I_{syn,i}^{\alpha}\), respectively, and \(\eta_i\) is a Gaussian white noise process with \(\delta\)-autocorrelation. After this diffusion approximation, in the mean-field limit \(N \rightarrow \infty\) we can describe the model equations ((1)-(3)) by two delay-coupled Fokker-Planck equations

$$ \frac{\partial p_{\alpha}}{\partial t} + \frac{\partial S_{\alpha}^V}{\partial V} + \frac{\partial S_{\alpha}^w}{\partial w} = 0 \tag{8}$$

where

$$ S_{\alpha}^V := \left(\frac{I_{ion}(V) - w + \mu_{\alpha}}{C} - \frac{\sigma_{\alpha} \partial \sigma_{\alpha}}{2C^2 \partial V}\right) p_{\alpha} - \frac{\sigma_{\alpha}^2 \partial p_{\alpha}}{2C^2 \partial V} \tag{9}$$ 

and

$$ S_{\alpha}^w := \frac{a(V-E_L) - w}{\tau_w} p_{\alpha} \tag{10}$$

Here,  \(p_{\alpha}(V,w,t)\) is the probability density to find a neuron belonging to population \(\alpha\) to be in a state \((V,w)\) at a given time \(t\). Furthermore, \(S_{\alpha}^V\) and \(S_{\alpha}^w\) are the probability fluxes in positive direction \(V\) and \(w\), respectively. The firing rate for population \(\alpha\) is then given by

$$ r_{\alpha}(t) := \int_R S_{\alpha}^V (V_{cut},w,t) dw \tag{11}$$

The described mean-field reduction yields a 2+1-dimensional PDE. Numerically solving the PDE is, however, still computationally expensive. Therefore, the system will be further reduced by an adiabatic approximation. Under the assumption that the timescales of the membrane potential and the adaptation current separable (see [Augustin et al., 2013] for details), the individual adaptation current \(w_i^{\alpha}(t)\) can be approximated by the population average \(w_{\alpha}\), evolving according to

$$ \tau_w \frac{d w_{\alpha}}{dt} = a((V)_{p_{\alpha (V,t)}}-E_L) - w_{\alpha} + \tau_w b r_{\alpha}(t) \tag{12}$$

where \((.)_p\) is the average over the density \(p\). This reduces the system to

$$ \frac{\partial p_{\alpha}}{\partial t} + \frac{\partial S_{\alpha}^V}{\partial V} = 0  \tag{13}$$

And again, the spike rate of a population \(\alpha\) is given by

$$ r_{\alpha}(t) := S_{\alpha}^V (V_{cut},t)  \tag{14}$$

This approach, however, is still computationally demanding because of the infinite state-space of the model. From that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations can be derived. This is achieved using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated [Augustin et al., 2017].

An evaluation of these reduced models finds that both approximation approaches lead to spike rate models that accurately reproduce the spiking behaviour of the underlying IF population. The low-dimensional models also well reproduce stable oscillatory spike rate dynamics that are generated either by recurrent synaptic excitation and neuronal adaptation or through delayed inhibitory synaptic feedback (Figure 2; taken from [Cakan and Obermayer, 2019]). The computational demands of the reduced models are very low. The derived spike rate descriptions retain a direct link to the properties of single neurons, allow for convenient mathematical analyses of network states, and are well suited for application in neural mass/mean-field based brain network models [Augustin et al., 2017; Cakan and Obermayer 2019].

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).
Lupe

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

[Augustin et al., 2013] Augustin, M., Ladenbauer, J., and Obermayer, K. (2013). How adaptation shapes spike rate oscillations in recurrent neuronal networks. Frontiers in Computational Neuroscience.

[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.

[Naud et al., 2008] Naud, R., Marcille, N., Clopath, C., and Gerstner, W. (2008). Firing patterns in the adaptive exponential integrate-and-fire model. Biological Cybernetics, 99(4-5), 335.

[Touboul and Brette, 2008] Touboul, J., and Brette, R. (2008). Dynamics and bifurcations of the adaptive exponential integrate-and-fire model. Biological Cybernetics, 99(4-5):319.

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