Highly connected neurons spike less frequently in balanced networks

Biological neuronal networks exhibit highly variable spiking activity. Balanced networks offer a parsimonious model of this variability in which strong excitatory synaptic inputs are canceled by strong inhibitory inputs on average, and irregular spiking activity is driven by fluctuating synaptic currents. Most previous studies of balanced networks assume a homogeneous or distance-dependent connectivity structure, but connectivity in biological cortical networks is more intricate. We use a heterogeneous mean-field theory of balanced networks to show that heterogeneous in-degrees can break balance. Moreover, heterogeneous architectures that achieve balance promote lower firing rates in neurons with larger in-degrees, consistent with some recent experimental observations.

Figure 1

A homogeneous balanced network. (a) Network schematic. A population of $N_{\mathrm{e}}$ excitatory and $N_{\mathrm{i}}$ inhibitory neurons ($\mathrm{e}$ and $\mathrm{i}$) are randomly connected and also receive feedforward input ($F_{\mathrm{e}}$ and $F_{\mathrm{i}}$). (b) Raster plot of 500 randomly sampled excitatory neurons from a simulation of a balanced network with $N_{\mathrm{e}}=4\times {10}^4$ and $N_{\mathrm{i}}={10}^4$. (c) Firing rates from simulations (solid curves) approach the values predicted by solving Eq. (3) (dashed lines) as network size, $N=N_{\mathrm{e}}+N_{\mathrm{i}}$, grows. (d) Synaptic input to one representative excitatory neuron shows that strong excitatory currents (blue) balance with strong inhibitory currents (red) to yield a moderate total synaptic current (black). Synaptic currents were convolved with a Gaussian-shaped filter ($\sigma =8$ ms) and normalized by the neuron's rheobase.

Figure 2

Heterogeneous in-degrees can break balance. (a) Network diagram. Same as the network in Fig. 1 except excitatory and inhibitory populations were each split into two populations. Neurons in populations ${\mathrm{e}}_2$ and ${\mathrm{i}}_2$ have larger in-degrees than those in ${\mathrm{e}}_1$ and ${\mathrm{i}}_1$. (b) Raster plot of 500 randomly selected excitatory neurons, half from ${\mathrm{e}}_1$ and half from ${\mathrm{e}}_2$, from a simulation with $N=5\times {10}^4$ neurons. (c),(d) Mean firing rate in each population as a function of network size ($N$).

Figure 3

Balance can be restored by heterogeneous out-degrees. Same as Fig. 2, except out-degrees of neurons in populations ${\mathrm{e}}_2$ and ${\mathrm{i}}_2$ were increased by rewiring a proportion $c_{\mathrm{out}}=4/5$ of the outgoing projections from populations ${\mathrm{e}}_1$ and ${\mathrm{i}}_1$ to project from ${\mathrm{e}}_2$ and ${\mathrm{i}}_2$ instead. Dashed lines show the asymptotic firing rates predicted by Eq. (3).

Figure 4

Dependence of firing rates on in-degree in a scale-free network. (a) Raster plot and (b) firing rates as a function of in-degree from a network of $5\times {10}^4$ neurons with a power-law distribution of in-degrees. For the raster plot, 500 excitatory neurons were sampled uniformly from the network and sorted so that in-degree increased with “neuron index.”