White-noise susceptibility and critical slowing in neurons near spiking threshold

We present mathematical and simulation analyses of the below-threshold noisy response of two biophysically motivated models for excitable membrane due to H. R. Wilson: a squid axon ("resonator") and a human cortical neuron ("integrator"). When stimulated with a low-intensity white noise superimposed on a dc control current, both membrane types generate voltage fluctuations that exhibit critical slowing down--that is, the voltage responsiveness to noisy input currents grows in amplitude while slowing in frequency--as the membrane approaches spiking threshold from below. We define threshold unambiguously as that dc current that renders a zero real eigenvalue for the Jacobian matrix for the integrator neuron, and, for the resonator neuron, as the dc current that gives a complex eigenvalue pair whose real part is zero. Using a linear Ornstein-Uhlenbeck analysis, we give exact small-noise expressions for the variance, power spectrum, and correlation function of the voltage fluctuations, and we derive the scaling laws for the divergence of susceptibility and correlation times for approach to threshold. We compare these predictions with numerical simulations of the nonlinear stochastic equations, and demonstrate that, provided the white-noise perturbations are kept sufficiently small, the linearized theory works well. These predictions should be testable in the laboratory using a current-clamped cell configuration. If confirmed, then the proximity of a neuron to its spike-transition point can be judged by measuring its subthreshold susceptibility to white-noise stimulation. We postulate that such temporally correlated fluctuations could provide a means of subthreshold signaling via gap-junction connections with neighboring neurons.