EI-Balance Network.

[NorbertZheng]

Related Reference

• Softky W R, Koch C. The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs.
• Shadlen M N, Newsome W T. Noise, neural codes and cortical organization.
• Xue M, Atallah B V, Scanziani M. Equalizing excitation–inhibition ratios across visual cortical neurons.
• Van Vreeswijk C, Sompolinsky H. Chaos in neuronal networks with balanced excitatory and inhibitory activity.
• Van Vreeswijk C, Sompolinsky H. Chaotic balanced state in a model of cortical circuits.
• Huang L, Cui Y, Zhang D, et al. Impact of noise structure and network topology on tracking speed of neural networks.

[NorbertZheng]

Historical Note

Rate Coding vs. Temporal Coding

• Rate Coding: Information is conveyed via the average rate of spikes.
• Temporal Coding: The precise timing of spikes conveys information.

Integrate-and-fire neurons unable to replicate irregular firing of cortical cells.

Softky et. al. find that

• Independent, random EPSPs (Excitatory Post-Synaptic Potential) include highly regular firing patterns, inconsistent with experimental findings.
• Neurons are coincidence detectors.

A balance of excitation and inhibition yields an irregular firing pattern.

Shadlen et. al. find that

• Membrane potential undergoes a random walk under balanced EPSPs (Excitatory Post-Synaptic Potential) and IPSPs (Inhibitory Post-Synaptic Potential).
• Irregular ISI (Inter-Spike Interval) can be modeled with the drift-diffusion process.
• Coincidence detector model would need $\tau\sim1ms$ to avoid the accumulation of EPSP (Excitatory Post-Synaptic Potential) on membrane potential.
• Drift-diffusion model is more biologically plausible.

[NorbertZheng]

Biological Evidence

Excitatory synapses outnumber inhibitory synapses by 6:1, but inhibitory synapses/neurons have

• more advantageous position, i.e. near soma
• larger conductances and longer durations
• higher firing rates

Direct evidence is found by Xue et. al.

• IPSCs (Inhibitory Post-Synaptic Currents) of PV neurons closely match EPSCs (Excitatory Post-Synaptic Currents) both temporally and spatially due to fast synaptic scaling of pyramidal cells.

[NorbertZheng]

EI-Balance Network

Based on Shadlen et. al., Vreeswijk et. al. investigates whether needs fine-tuning of parameters and computational benefits.

Neuron Dynamics

$\Theta(.)$ is Heaviside function.

$$ \sigma{k}^{i}(t)=\Theta\left(u{k}^{i}(t)\right) $$

$$ u{k}^{i}(t)=\Sigma{l=1}^{2}\Sigma{j=1}^{N{l}}J{kl}^{ij}\sigma{l}^{j}(t)+u{k}^{0}-\theta{k} $$

$$ J{EE}=J{IE}=1; J{E}=-J{EI}; J{I}=-J{II} $$

Mean activity $m_{k}^{i}(t)$

$$ m{k}^{i}(t)=\left<\sigma{k}^{i}(t)\right> $$

Mean-field results

• Linear encoding property resulted from the linear summation of E\&I current, regardless of channel and activation non-linearity (i.e. works for Hodgkin-Huxley model as well).

$$ u{E}=(Em{0}+m{E}-J{E}m{I})\sqrt{K}-\theta_{E} $$

$$ u{I}=(Im{0}+m{E}-J{I}m{I})\sqrt{K}-\theta{I} $$

Balanced state necessary condition: $0 < m < 1$ even when $K$ is large.

$$ Em{0}+m{E}-J{E}m{I}=O(\frac{1}{\sqrt{K}}) $$

$$ Im{0}+m{E}-J{I}m{I}=O(\frac{1}{\sqrt{K}}) $$

Then, we get

$$ m{E}=\frac{J{I}E-J{E}I}{J{E}-J{I}}m{0}=A{E}m{0} $$

$$ m{I}=\frac{E-I}{J{E}-J{I}}m{0}=A{I}m{0} $$

And the requirements are

$$ m{E}>0; m{I} > 0 $$

$$ \frac{E}{I}>\frac{J{E}}{J{I}}>1 $$

$$ J_{E}>1 $$

[NorbertZheng]

Mean-field analysis of stable stationary population firing rates $m_{k}^{i}(t)$

Dynamics (Ginzhurg \& Sompolinsky, 1994)

$$ \tau{k}\frac{d}{dt}m{k}^{i}(t)=-m{k}^{i}(t)+\Theta\left(u{k}^{i}(t)\right) $$

Rewriting $u{k}^{i}(t)$ , where $p{l}(n{l})$ is the probability of receiving $n{l}$ spikes from population $l$.

Now, we get the following conclusions

• Fixed point stability.
• Fast response to the external changing stimulus.

[NorbertZheng]

On the Fast Response Property

Huang et. al. find that

• Structured noise can also facilitate network response.

[NorbertZheng]

More on the Sparse Assumption

[NorbertZheng]

Application

Parametric spatial working memory

Mitigate unreliability in Poisson coding