Firing rate dynamics in recurrent spiking neural networks with intrinsic and network heterogeneity

Abstract

Heterogeneity of neural attributes has recently gained a lot of attention and is increasing recognized as a crucial feature in neural processing. Despite its importance, this physiological feature has traditionally been neglected in theoretical studies of cortical neural networks. Thus, there is still a lot unknown about the consequences of cellular and circuit heterogeneity in spiking neural networks. In particular, combining network or synaptic heterogeneity and intrinsic heterogeneity has yet to be considered systematically despite the fact that both are known to exist and likely have significant roles in neural network dynamics. In a canonical recurrent spiking neural network model, we study how these two forms of heterogeneity lead to different distributions of excitatory firing rates. To analytically characterize how these types of heterogeneities affect the network, we employ a dimension reduction method that relies on a combination of Monte Carlo simulations and probability density function equations. We find that the relationship between intrinsic and network heterogeneity has a strong effect on the overall level of heterogeneity of the firing rates. Specifically, this relationship can lead to amplification or attenuation of firing rate heterogeneity, and these effects depend on whether the recurrent network is firing asynchronously or rhythmically firing. These observations are captured with the aforementioned reduction method, and furthermore simpler analytic descriptions based on this dimension reduction method are developed. The final analytic descriptions provide compact and descriptive formulas for how the relationship between intrinsic and network heterogeneity determines the firing rate heterogeneity dynamics in various settings.

Appendix

Given two vectors 𝜃 (intrinsic heterogeneity) and q (network heterogeneity), we can generate a new pair of vectors (of the same size) that have any desired correlation coefficient ϱ∈(−1,1). In this paper, we choose to keep q fixed and generate a new vector 𝜗 that has the same sample mean (μ(𝜃)) and sample standard deviation (σ(𝜃)) of 𝜃. Note that there are infinitely many ways to generate two such vectors if we only require that the mean and standard deviation of the new vectors be equal to the original statistics of the vector. The algorithm we use is as follows.

• INPUTS: (q, 𝜃, ϱ)

• Set φ= cos−1(ϱ)

• Shift input vectors so they have zero mean:

  q ₀ = q−μ(q)

  𝜃 ₀ = 𝜃−μ(𝜃).

• Calculate orthogonal complement to q ₀:

  $\mathbf {z}=\boldsymbol {\theta }_{0}-\frac {\mathbf {q}_{0}\cdot \boldsymbol {\theta }_{0}}{\|\mathbf {q}_{0}\|^{2}}\mathbf {q}_{0}$

• Create unit vectors out of q ₀ and z:

  $\tilde {\mathbf {q}}=\mathbf {q}_{0}/\|\mathbf {q}_{0}\|$, $\tilde {\mathbf {z}}=\mathbf {z}/\|\mathbf {z}\|$

• Create vector with prescribed correlation and zero mean:

  $\hat {\boldsymbol {\theta }}=\cos (\varphi )\tilde {\mathbf {q}}+\sin (\varphi )\tilde {\mathbf {z}}$

• Set $\boldsymbol {\vartheta }=\frac {\sigma (\boldsymbol {\theta })}{\sigma (\hat {\boldsymbol {\theta }})}\hat {\boldsymbol {\theta }}+\mu (\boldsymbol {\theta })$

• OUTPUT: 𝜗, where correlation coefficient of 𝜗 and q is ϱ, μ(𝜗) = μ(𝜃), and σ(𝜗) = σ(𝜃).