Manuscript (handwritten)

Eoin Murray and Timothy O’Leary.

The PING-gated-sparsity paper. This entry is the methods backbone — neuron model, network architecture, training recipe, and mean-field derivation — kept in one place so the results-and-discussion companion ar010 and the notebook chain in the Gamma Gated Sparsity collection can cite a single definitions source. The aggregation companion (ar010) carries the introduction, results figures, discussion, and conclusion.

Methods

The neuron model

This paper uses a conductance-based leaky integrate-and-fire (LIF) network with separate excitatory and inhibitory populations, fixed recurrent E↔I matrices, and exponential synapses. The COBA neurons obey

where $V$ is membrane potential, $t$ is time, $C_m$ is capacitance, $g_L$ is leak conductance, $g_e$ is the excitatory synaptic conductance and $g_i$ is the inhibitory synaptic conductance with reversal potentials $E_L = -65~\text{mV}$ (leak), $E_e = 0~\text{mV}$ (excitatory), $E_i = -80~\text{mV}$ (inhibitory). Each conductance term pulls $V$ toward its reversal potential at a rate proportional to the conductance. Excitatory synapses drive $V$ up towards $E_e = 0$ and inhibitory synapses both pull it down towards $E_i = -80$ and increase total conductance, which lowers the effective membrane time constant — this is shunting inhibition.

Equation (1) can be written as a linear first-order ODE in the form $C_m \dot{V} = -AV + B$ with $A = g_L + g_e + g_i$ and $B = g_L E_L + g_e E_e + g_i E_i$. Setting $\dot{V} = 0$ we find the voltage at which the inflow $B$ exactly balances the outflow $AV$, and name that voltage $V_\infty$. We define accordingly:

Substituting equations (2)–(4) into (1) and dividing through by $g_{\text{tot}}$ yields the standard “decay-to-steady-state” form

Integrating over an interval where $\tau_{\text{eff}}$ and $V_\infty$ are held constant gives the closed-form solution

the membrane interpolates from its starting value $V_0$ toward $V_\infty$ on timescale $\tau_{\text{eff}}$. However in continuous time $\tau_{\text{eff}}$ and $V_\infty$ are not constant — both depend on $g_e(t)$ and $g_i(t)$ which evolve as spikes arrive. A zero-order hold makes this numerically tractable: freeze $g_e$ and $g_i$ over one timestep $\Delta t$, integrate (5) exactly under that freeze, then update $g_e$ and $g_i$ for the next timestep. Under that freeze, (5) integrates analytically to

After the membrane update, apply the spike reset:

with $V_{\text{th}} = -50~\text{mV}$ and $V_{\text{reset}} = -65~\text{mV}$. Refractory periods are $\tau^e_{\text{ref}} = 3~\text{ms}$ and $\tau^i_{\text{ref}} = 1.5~\text{ms}$, held in a per-neuron countdown that suppresses spiking until exhaustion.

Each conductance is the convolution of its presynaptic spike train with a decaying exponential. For a single neuron receiving feedforward input only:

where $\alpha = \exp(-\Delta t / \tau_{\text{AMPA}})$ is the per-step AMPA decay factor. Between spikes the conductance decays exponentially toward zero; each arriving spike adds a kick proportional to the synaptic weight. The inhibitory conductance $g_i$ follows the same form with GABA decay $\gamma = \exp(-\Delta t / \tau_{\text{GABA}})$.

In a network with E and I populations, two recurrent pathways close the PING loop alongside the feedforward input. There are no E→E connections.

Equation (6) is the excitatory conductance on E neurons — purely feedforward, no recurrent E→E term. Equation (7) is the inhibitory conductance on E neurons — driven by I spikes through $W_{\text{ie}}$. Equation (8) is the excitatory conductance on I neurons — driven by E spikes through $W_{\text{ei}}$. I neurons receive no inhibition. The trainable feedforward input $W_{\text{in}} s^{\text{inp}}_t$ enters E’s excitatory conductance after the decay step, arriving fresh each timestep regardless of recurrent state.

The model emits two modes, COBA and PING; when PING mode is active the $W_{\text{ei}}$ and $W_{\text{ie}}$ matrices are fixed at init and are not trainable. They are controlled by a single scalar flag $s$:

At $s = 0$ both matrices are zero, the loop is open, and the network is feedforward COBA. At $s = 1$ the default initialisation gives $W_{\text{ei}} \approx 1~\mu\text{S}$ and $W_{\text{ie}} \approx 2~\mu\text{S}$, strong enough to sustain the PING rhythm from the first forward pass of training.

For each timestep $t \to t+1$, the simulator applies the four updates below in a fixed order. The two populations are updated by the same template but with different driving inputs; they’re written out separately here so the recurrent coupling is explicit.

The E and I populations are updated synchronously: at step $t+1$ both read the spike outputs $s^e_t$, $s^i_t$ from step $t$ to drive their conductances, and both emit fresh spikes $s^e_{t+1}$, $s^i_{t+1}$ at the end of the step. Synchronous-step semantics keep the E→I→E gamma cycle phase-locked to the integration grid.

Network model

Training

Mean field derivation

Homogeneous coupling

The weight matrices in the simulator are random with finite variance. We replace them by their means: $W^{EI}_{kj} \to w^{EI}$ and $W^{IE}_{jk} \to w^{IE}$ for all $j, k$, and introduce population mean firing rates

so that the presynaptic summation terms in equations (6)–(8) collapse onto these rates:

Under equations (10)–(11) every E cell sees the same $g_i^E$ and every I cell sees the same $g_e^I$. The per-cell conductance equations collapse to two population-mean equations.

Defining lumped couplings

the conductance dynamics become

Driving force linearisation

The synaptic terms in equation (1) still depend on $V$ through driving force terms $(V - E_i)$ and $(V - E_e)$. Take $V$ to its resting value in the synaptic terms, with $V_\text{rest} = -65$ mV:

(so $-(V - E_e) \approx -\Delta V_\text{exc} = +65$ mV — depolarising drive on I). The synaptic currents in equation (1) become

each synaptic input is now linear in conductance.

Population rate from f-I curve

A LIF cell receiving constant net input current $I$ fires at a rate $\phi(I)$ where $\phi$ is its f-I curve. Replacing each cell’s spike output by its rate response gives:

with effective input currents (from equation (1) with the linearisation (17)–(18) substituted):

The instantaneous-rate replacement (19) is only valid when input changes are slow compared to the membrane integration time. In reality, $E(t)$ relaxes toward the f-I-curve fixed point on its membrane time constant $\tau_E$. Encode this with an explicit relaxation:

where $\Phi_E, \Phi_I$ are the smooth steady-state gain functions (sigmoids in the numerics). Drives that change faster than $\tau_E, \tau_I$ are filtered; the rate doesn’t snap to the f-I curve instantaneously. Together with the conductance dynamics (13)–(14) we now have four equations in $(E, I, g_e^I, g_i^E)$.

Absorb the driving-force constants

The fixed prefactors $\Delta V_\text{inh}, |\Delta V_\text{exc}|$ in equations (22)–(23) and the fan-in scalings $\tilde W^{EI}, \tilde W^{IE}$ in (12) are just multiplicative constants — they don’t carry dynamical information. Fold them into the couplings:

and likewise absorb the $|\Delta V_\text{exc}|$ inside the I-cell argument by redefining $g_e^I \mapsto g_e^I \cdot |\Delta V_\text{exc}|$ (similarly for $g_i^E$). The conductances now have units of input current rather than µS; the f-I curves take their argument directly. This change of variables is pure bookkeeping — no dynamics are lost.

The 4D system

After applying equations (10)–(24), the mean-field system in state $(E, I, g_e^I, g_i^E)$ is

Equations (25)–(26) are the canonical 4D Wilson-Cowan-type reduction.

Summary of approximations

4D Jacobian and Hopf criterion

At a fixed point $(E^\star, I^\star, g_e^{I\star}, g_i^{E\star})$:

with $\Phi'_E, \Phi'_I$ at the fixed-point arguments. The Hopf is the smallest $I_\text{ext}^\star$ at which the leading pair crosses the imaginary axis: $\mathrm{Re}(\lambda) = 0$, $\mathrm{Im}(\lambda) = \omega^\star$, with predicted gamma frequency $f^\star = \omega^\star / (2\pi)$.