Mean-field Model

The Model

Shared and private sources have constant rates and enter through stationary mean conductances. Their filtering transients and covariance are omitted.

States

Six states: two population rates and four recurrent conductances.

Rates

$r_E,r_I$ are firing rates; $\tau_{r,E},\tau_{r,I}>0$ are relaxation times. $\Phi_E,\Phi_I$ are the stationary noisy-LIF gains in Equation 15; $I_E,I_I$ are defined in Equations 9–10.

Recurrent conductances

$g_e^Q,g_i^Q$ are mean excitatory and inhibitory conductances into population $Q\in\{E,I\}$. For pathway $P\to Q$, $j$ indexes $N_P$ sources and $k$ indexes $N_Q$ targets. Each $\mathbf W_{jk}^{P\to Q}\ge0$ is a conductance jump per spike; absent connections have zero weight. Dividing by target size $N_Q$ averages incoming drive, assuming source rate $r_P$ is homogeneous. $\tau_{P\to Q}>0$ is the synaptic decay time.

External input, effective currents and state count

All $M$ shared sources have rate $r_X$; $a$ indexes sources and $k$ targets. Dividing each weight sum by target population size gives mean incoming weight. $\tau_{\mathrm{AMPA},X}>0$ is the conductance decay time.

Private populations $X_E,X_I$ provide one independent source per neuron, all at rate $r_{\mathrm{priv}}$, which may differ from $r_X$. Their diagonal matrices connect source $k$ only to target $k$; the sums average these weights over targets. $\tau_{\mathrm{AMPA},\mathrm{priv}}>0$ is the decay time.

$I_{\mathrm{ext},E},I_{\mathrm{ext},I}$ are applied currents. Fixed driving forces are $\Delta V_{\mathrm{exc},Q}=E_e-E_{L,Q}$ and $\Delta V_{\mathrm{inh},Q}=E_{L,Q}-E_i$, where $E_e,E_i$ are synaptic reversal potentials and $E_{L,Q}$ is the leak reversal potential (resting voltage without other input).

With $\mathbf W^{E\to E}=\mathbf W^{I\to I}=0$ and $g_e^E(0)=g_i^I(0)=0$, both self-conductances remain zero. The model then has four states: $(r_E,r_I,g_e^I,g_i^E)$.

Units: ms for time $t$, spikes/neuron/ms for rates, µS for conductances and weights, mV for voltages, and nA for currents. A dot denotes $d/dt$.

Derivation

From spikes to mean conductances

For a pathway from population $P$ to $Q$, each source spike adds conductance to target neuron $k$:

Here $g_{P\to Q,k}$ is conductance, $\tau_{P\to Q}$ its decay time, $s_j^P(t)$ source neuron $j$‘s spike train, and $\mathbf W^{P\to Q}_{jk}$ the conductance jump per spike. A dot denotes a time derivative.

Taking expectations, assuming a common source rate $r_P(t)$, and averaging over the $N_Q$ targets gives

where $g_{P\to Q}$ is population-average expected conductance. Expectation is exact for fixed weights; homogeneous source rates are an approximation.

External conductances follow the same equation, with constant shared rate $r_X$ and private rate $r_{\mathrm{priv}}$. Setting their derivatives to zero gives Equations 7–8 after adding shared and private input. This eliminates four external states when their initial means are stationary. Changing rates or nonstationary initial means require restoring the input filters; instantaneous stationary substitution would be an additional approximation. Input covariance is omitted throughout.

From conductances to effective currents

A conductance $g$ with reversal potential $E_{\mathrm{rev}}$ produces current

where $V_m$ is membrane voltage. Approximate the driving force at the leak reversal potential $E_{L,Q}$ of population $Q$, defining $\Delta V_{\mathrm{exc},Q}=E_e-E_{L,Q}$ and $\Delta V_{\mathrm{inh},Q}=E_{L,Q}-E_i$. Here $E_e,E_i$ are excitatory and inhibitory reversal potentials.

Adding recurrent, shared and private contributions gives

Here $I_{\mathrm{ext},Q}$ is applied current; $g_e^Q,g_i^Q$ are recurrent conductances; and $g_{\mathrm{ext}}^Q$ is total stationary external excitatory conductance, combining shared and private input.

Fixing the driving forces removes voltage dependence, omitting shunting and conductance-dependent membrane time constants.

From effective currents to firing rates

For fixed current $I_Q$, approximate unresolved fluctuations by Gaussian white noise. The stationary firing rate is the inverse of the mean time to reach threshold plus the refractory period. The Siegert formula[1] gives

with

Here $g_{L,Q}$ is leak conductance, $\tau_{m,Q}=C_{m,Q}/g_{L,Q}$ is the membrane time constant, and $C_{m,Q}$ is capacitance. The voltages $V_{\mathrm{reset},Q},V_{\mathrm{th},Q}$ specify reset and threshold; $\tau_{\mathrm{ref},Q}$ is the refractory period. The fixed parameter $\sigma_{V,Q}>0$ sets the voltage-noise scale; $u$ is dimensionless and $\operatorname{erf}$ is the error function.

To give the population a finite response time, let its rate relax toward this stationary gain:

where $r_Q$ is firing rate and $\tau_{r,Q}>0$ is the relaxation time. This relaxation law is a closure assumption; the noise scale is prescribed rather than derived from network fluctuations.

Gain slope

For Jacobian calculations, differentiate Equation 15 with respect to current, holding the noise scale fixed. Define

The Leibniz rule gives

Here $\Phi_Q'$ is the gain’s current derivative; the remaining terms are defined above.

Four-state reductions

Zero E→E and I→I weights with zero initial self-conductances give the four states $(r_E,r_I,g_e^I,g_i^E)$ directly. Nonzero self-coupling also permits four states when outgoing E pathways share a decay time, outgoing I pathways share a decay time, and initial mean conductances satisfy

The cross-pathway sums must be nonzero. Subtracting each reconstructed conductance from its full filter equation leaves a difference that decays exponentially; it remains zero when initially zero. Arbitrary initial differences or unequal outgoing decay times require the additional states. These identities apply to population means, not individual neuronal conductances. External input remains parameterised as above.

Stability Analysis

Equilibrium

Hold source rates, weights and applied currents fixed. An equilibrium $x^*$ satisfies $F(x^*)=0$, where $\dot x=F(x)$ denotes the six-state model. Setting each recurrent conductance derivative to zero gives

Substitute these conductances and the stationary external inputs into the current definitions, then solve the two rate conditions

A star denotes equilibrium. Here $g_{E\to Q}=g_e^Q$ and $g_{I\to Q}=g_i^Q$. Multiple equilibria may exist; analyse each separately.

Analytical Jacobian

For a small perturbation $\delta x=x-x^*$, the first-order dynamics are

To keep the matrix readable, define the excitatory and inhibitory rate sensitivities for target population $Q$:

The gain slope $\Phi_Q'$ is given in Equation 19. Using the state order above, the full Jacobian is

Each entry is the derivative of one state’s time derivative with respect to another state. The first two rows describe conductance-to-rate effects; the last four describe rate-to-conductance feedback and synaptic decay. External input has no dynamical rows, but changes the equilibrium and hence $A_Q,B_Q$.

Stability and oscillatory onset

Calculate the six eigenvalues $\lambda_J$ of $J_{\mathrm{flow}}$. Negative real parts for every eigenvalue imply local asymptotic stability; any positive real part implies instability. If any real part is zero and none is positive, linearisation alone is inconclusive.[2]

A complex eigenvalue describes a linear oscillatory mode with frequency

where $\operatorname{Im}\lambda_J$ is its imaginary part in radians/ms. A complex pair crossing the imaginary axis at nonzero frequency, with the other modes stable and a nonzero crossing slope, identifies a Hopf candidate. Nonlinear nondegeneracy and criticality require further analysis; the Jacobian does not determine the stability or amplitude of an emerging cycle.[2]

References

1. K. Kreutz-Delgado. “Mean Time-to-Fire for the Noisy LIF Neuron: A Detailed Derivation of the Siegert Formula.” arXiv (2015). doi:10.48550/arXiv.1501.04032

2. Y. A. Kuznetsov. Elements of Applied Bifurcation Theory, third edition. Springer (2004), Chapters 3 and 5. doi:10.1007/978-1-4757-3978-7