COBANet

COBANet#

ar003 · 14 May 2026 · pdf

1. A conductance based neuron equation#

The membrane is a capacitor ( $𝐶_{𝑚}$ ) pierced by ion channels in parallel. Conservation of charge (Kirchhoff) balances the capacitive current $𝐶_{𝑚} d 𝑉 / d 𝑡$ against the total ionic current:

𝐶_{𝑚} \frac{d 𝑉}{d 𝑡} = - \sum_{ion} 𝐼_{ion} (1)

Each channel passes an ohmic current — its conductance $𝑔_{ion} \geq 0$ times the driving force $(𝑉 - 𝐸_{ion})$ , the distance of $𝑉$ from the reversal potential $𝐸_{ion}$ (where the channel’s net current vanishes, set by the Nernst equilibrium):

𝐼_{ion} = 𝑔_{ion} (𝑉 - 𝐸_{ion}) (2)

Because $𝑔_{ion} \geq 0$ , the current’s sign lives entirely in the driving force. Summing a leak ( $𝑔_{𝐿}$ , $𝐸_{𝐿}$ ) and synaptic conductances — excitatory ( $𝑔_{𝑒}$ , $𝐸_{𝑒}$ ), inhibitory ( $𝑔_{𝑖}$ , $𝐸_{𝑖}$ ) — gives the general conductance-based (COBA) neuron:

𝐶_{𝑚} \frac{d 𝑉}{d 𝑡} = - 𝑔_{𝐿} (𝑉 - 𝐸_{𝐿}) - 𝑔_{𝑒} (𝑉 - 𝐸_{𝑒}) - 𝑔_{𝑖} (𝑉 - 𝐸_{𝑖}) (3)

2. The COBA model#

COBANet specialises (3) to two populations — E driven by excitation and inhibition, I by excitation only:

𝐶_{𝑚}^{𝐸} \frac{d 𝑉^{𝐸}}{d 𝑡} = - 𝑔_{𝐿}^{𝐸} (𝑉^{𝐸} - 𝐸_{𝐿}) - 𝑔_{𝑒}^{𝐸} (𝑉^{𝐸} - 𝐸_{𝑒}) - 𝑔_{𝑖}^{𝐸} (𝑉^{𝐸} - 𝐸_{𝑖}) (4)

𝐶_{𝑚}^{𝐼} \frac{d 𝑉^{𝐼}}{d 𝑡} = - 𝑔_{𝐿}^{𝐼} (𝑉^{𝐼} - 𝐸_{𝐿}) - 𝑔_{𝑒}^{𝐼} (𝑉^{𝐼} - 𝐸_{𝑒}) (5)

A neuron spikes at threshold $𝑉_{th}$ and resets to $𝑉_{reset}$ for a refractory period:

𝑠_{𝑡 + 1} = 𝟙 [𝑉 \geq 𝑉_{th}], 𝑉 \leftarrow 𝑉_{reset} if 𝑠_{𝑡 + 1} = 1 or refractory (6)

Each synaptic conductance is an exponential trace driven by presynaptic spikes — each spike adds its full weight as an instantaneous jump, then the conductance decays with the channel time constant; there is no E→E connection:

\frac{d 𝑔_{𝑒}^{𝐸}}{d 𝑡} = - \frac{𝑔_{𝑒}^{𝐸}}{𝜏_{AMPA}} + 𝑊_{in} \sum_{𝑘} 𝛿 (𝑡 - 𝑡_{𝑘}^{inp}) (7)

\frac{d 𝑔_{𝑖}^{𝐸}}{d 𝑡} = - \frac{𝑔_{𝑖}^{𝐸}}{𝜏_{GABA}} + 𝑊_{ie} \sum_{𝑘} 𝛿 (𝑡 - 𝑡_{𝑘}^{𝑖}) (8)

\frac{d 𝑔_{𝑒}^{𝐼}}{d 𝑡} = - \frac{𝑔_{𝑒}^{𝐼}}{𝜏_{AMPA}} + 𝑊_{ei} \sum_{𝑘} 𝛿 (𝑡 - 𝑡_{𝑘}^{𝑒}) (9)

(7) is E’s excitation from the input $𝑊_{in}$ ; (8) its inhibition from I via $𝑊_{ie}$ ; (9) the I population’s excitation from E via $𝑊_{ei}$ .

3. Discretization#

The conductances (7)–(9) and membrane equations (4)–(5) are continuous ODEs. The delta-driven conductances integrate exactly over one step: between spikes they decay by $𝑒^{- Δ 𝑡 / 𝜏}$ , and any spike landing in the step adds its full weight — the decay-then-add recurrence $𝑔_{𝑡 + 1} = 𝑒^{- Δ 𝑡 / 𝜏} 𝑔_{𝑡} + 𝑊 𝑠_{𝑡}$ (with the $𝜏$ , $𝑊$ and spike train $𝑠$ of each of (7)–(9)). The membrane we integrate by exponential Euler — the same algebra for both populations (the I neuron drops $𝑔_{𝑖}$ ).

Collecting on $𝑉$ makes it linear, with total conductance $𝑔_{tot} = 𝑔_{𝐿} + 𝑔_{𝑒} + 𝑔_{𝑖}$ :

𝐶_{𝑚} \frac{d 𝑉}{d 𝑡} = - (𝑔_{𝐿} + 𝑔_{𝑒} + 𝑔_{𝑖}) 𝑉 + (𝑔_{𝐿} 𝐸_{𝐿} + 𝑔_{𝑒} 𝐸_{𝑒} + 𝑔_{𝑖} 𝐸_{𝑖}) (10)

Dividing by $𝑔_{tot}$ gives decay-to-steady-state form, naming $𝜏_{eff} = 𝐶_{𝑚} / 𝑔_{tot}$ (shorter than $𝐶_{𝑚} / 𝑔_{𝐿}$ when synapses are open) and the steady-state voltage $𝑉_{\infty}$ (the conductance-weighted mean of the reversals):

\frac{𝐶_{𝑚}}{𝑔_{tot}} \frac{d 𝑉}{d 𝑡} = - (𝑉 - \frac{𝑔_{𝐿} 𝐸_{𝐿} + 𝑔_{𝑒} 𝐸_{𝑒} + 𝑔_{𝑖} 𝐸_{𝑖}}{𝑔_{tot}}) (11)

A zero-order hold freezes the conductances over one step $Δ 𝑡$ , leaving (11) constant-coefficient with exact solution

𝑉_{𝑡 + 1} = 𝑉_{\infty} + (𝑉_{𝑡} - 𝑉_{\infty}) 𝑒^{- Δ 𝑡 / 𝜏_{eff}} (12)

Per population — I has no $𝑔_{𝑖}$ , so its $𝑔_{tot}$ and $𝑉_{\infty}$ drop those terms:

𝑔_{tot}^{𝐸} = 𝑔_{𝐿}^{𝐸} + 𝑔_{𝑒}^{𝐸} + 𝑔_{𝑖}^{𝐸}, 𝜏_{eff}^{𝐸} = \frac{𝐶_{𝑚}^{𝐸}}{𝑔_{tot}^{𝐸}}, 𝑉_{\infty}^{𝐸} = \frac{𝑔_{𝐿}^{𝐸} 𝐸_{𝐿} + 𝑔_{𝑒}^{𝐸} 𝐸_{𝑒} + 𝑔_{𝑖}^{𝐸} 𝐸_{𝑖}}{𝑔_{tot}^{𝐸}} (13)

𝑔_{tot}^{𝐼} = 𝑔_{𝐿}^{𝐼} + 𝑔_{𝑒}^{𝐼}, 𝜏_{eff}^{𝐼} = \frac{𝐶_{𝑚}^{𝐼}}{𝑔_{tot}^{𝐼}}, 𝑉_{\infty}^{𝐼} = \frac{𝑔_{𝐿}^{𝐼} 𝐸_{𝐿} + 𝑔_{𝑒}^{𝐼} 𝐸_{𝑒}}{𝑔_{tot}^{𝐼}} (14)

with step (12) for each population $𝑝 \in {𝐸, 𝐼}$ : $𝑉_{𝑡 + 1}^{𝑝} = 𝑉_{\infty}^{𝑝} + (𝑉_{𝑡}^{𝑝} - 𝑉_{\infty}^{𝑝}) 𝑒^{- Δ 𝑡 / 𝜏_{eff}^{𝑝}}$ .

Being the exact frozen-conductance integral, (12) is dt-invariant — $𝑁$ small steps equal one big step, so firing rates and the gamma frequency are physical (Hz) properties, not timestep artifacts (exp044). A forward-Euler step $𝑉_{𝑡 + 1} = 𝑉_{𝑡} + (Δ 𝑡 / 𝐶_{𝑚}) 𝐼_{net (𝑉_{𝑡})}$ — not dt-invariant — is kept only as a parity toggle (COBA_INTEGRATOR).

Each step runs in fixed order: conductances (7)–(9), then $𝑔_{tot}, 𝜏_{eff}, 𝑉_{\infty}$ , then the membrane step (12), then spike + reset (6). The zero-order hold is this ordering — the conductances advance once, then stay fixed while the membrane integrates across $Δ 𝑡$ . E and I advance synchronously, phase-locking the E→I→E gamma cycle to the grid.