COBANet

COBANet is a conductance-based leaky integrate-and-fire (LIF) spiking neural network with separate excitatory (E) and inhibitory (I) populations, fixed recurrent E↔I connectivity, and exponential synapses. This article derives the model equations and states the architectural constants. All parameter values follow Börgers (2017).

Neuron model

Membrane dynamics

Each neuron’s membrane potential $V$ evolves according to the conductance-based LIF equation:

where $C_m$ is capacitance, $g_L$ is leak conductance, $g_e$ and $g_i$ are the total excitatory and inhibitory synaptic conductances, and $E_L$, $E_e$, $E_i$ are the corresponding reversal potentials. Each term pulls $V$ toward its reversal at a rate proportional to the conductance.

Collecting on $V$:

Define the total conductance, effective time constant, and steady-state voltage:

Then (2) becomes:

Note that $\tau_{\text{eff}}$ is not the leak-only time constant $\tau_m = C_m / g_L$ — when synapses are open, $g_{\text{tot}} > g_L$ and the membrane responds faster. This is shunting inhibition: open $g_i$ increases $g_{\text{tot}}$, shortening $\tau_{\text{eff}}$ and dampening the membrane’s response to all inputs.

Discretisation

Hold $g_e$, $g_i$ constant over $[t, t + \Delta t]$ (zero-order hold). Then $\tau_{\text{eff}}$ and $V_\infty$ are constant and (4) integrates exactly:

Spike and reset

with $V_{\text{th}} = -50$ mV and $V_{\text{reset}} = -65$ mV. Reset is hard (overshoot discarded). After spiking the neuron enters a refractory period during which $V$ is clamped at $V_{\text{reset}}$ and no spike can be emitted.

Synaptic conductances

Each conductance is the convolution of its presynaptic spike train with a decaying exponential:

Between spikes the conductance decays exponentially toward zero; each presynaptic spike adds a kick proportional to the synaptic weight $W$. Excitatory synapses use the AMPA time constant ($\tau_{\text{AMPA}} = 2$ ms); inhibitory synapses use the GABA time constant ($\tau_{\text{GABA}} = 9$ ms).

Population structure

The network has a single hidden layer partitioned into E and I sub-populations in a 4:1 ratio. E and I neurons obey the same LIF equations (1)–(6) but with different biophysical constants:

Three asymmetries matter for the PING cycle:

1. I neurons integrate four times faster ($\tau_m^I = 5$ ms vs $\tau_m^E = 20$ ms). A synchronous E-burst raises I above threshold within a single AMPA window (≈2 ms).
2. GABA decay is 4.5× slower than AMPA ($\tau_{\text{GABA}} = 9$ ms vs $\tau_{\text{AMPA}} = 2$ ms). The resulting I→E inhibition outlasts the triggering excitation, suppressing E long enough for excitatory drive to re-accumulate. This sets the gamma period.
3. I neurons receive no inhibition. Only $g_e$ is wired into the I update. The I population follows E’s drive without self-suppression.

Recurrent connectivity

There are no E→E connections ($W^{EE} = 0$, Börgers-style PING). Two recurrent pathways close the loop alongside feedforward input:

Eq (8): excitatory conductance on E neurons, purely feedforward. Eq (9): inhibitory conductance on E neurons, driven by I spikes through $W^{IE}$. Eq (10): excitatory conductance on I neurons, driven by E spikes through $W^{EI}$.

$W^{EI}$ and $W^{IE}$ are controlled by the scalar —ei-strength $s$:

where $r$ is the —ei-ratio (default 2.0). At $s = 0$ both matrices are zero, the loop is open, and the network is feedforward COBA. At $s = 1$ the initialisation gives $W^{EI} \approx 1$ µS and $W^{IE} \approx 2$ µS — sufficient to sustain PING from the first forward pass. Both matrices are held requires_grad = False; the optimiser cannot modify them.

Trainable parameters

Only two weight matrices train:

• $W_{\text{in}}$ ($N_{\text{in}} \times N_E$): feedforward input to E conductance. Initialised from $\mathcal{N}(\mu, 0.1\mu)$ with 95% sparsity. Default $\mu$ depends on configuration: 0.3 for COBA, 1.2 for PING.
• $W_{\text{out}}$ ($N_E \times N_{\text{classes}}$): linear readout from the time-averaged E spike vector (mem-mean mode).

All recurrent weights ($W^{EI}$, $W^{IE}$), all biophysical constants ($\tau_m$, $C_m$, $g_L$, $\tau_{\text{AMPA}}$, $\tau_{\text{GABA}}$, reversal potentials), and the E→E matrix ($W^{EE} = 0$) are frozen.

Constants