Fig. 3: Stability of adLIF discretizations.
From: Advancing spatio-temporal processing through adaptation in spiking neural networks
a Membrane potential u(t) over time for an EF-adLIF (left) and a SE-adLIF (right) neuron for different discretization time steps Δt ∈ {0.001, 0.5, 1}. Both neurons have the same parameters (τu = 25 ms, τw = 60 ms, a = 120). b Relationship between the decay rate r and discretization time step Δt for adLIF models with different discretizations, EF and SE. All decay rates are calculated with respect to 1 ms, a decay rate of r = 0.9 hence represents a decrease in magnitude of 10% every 1 ms. The decay rate (r = 0.972) of the equivalently parameterized continuous model is highlighted. Same neuron parameters as in panel (a). c Intrinsic frequency f and per-timestep decay rate r of 1000 different parameterizations of adLIF neurons for Euler-Forward discretization (left), SE (right), and the continuous model (middle). The horizontal dotted line at r = 1.0 marks the stability bound. Instances above this line diverge due to exponential growth. Parameter ranges are uniformly distributed over the intervals a ∈ [0, 120], τu ∈ [5, 25] ms and τw ∈ [60, 300] ms. d Eigenvalues of \({\bar{A}}_{{{{\rm{EF}}}}}\) (left) and \({\bar{A}}_{{{{\rm{SE}}}}}\) (right) plotted in the complex plane for fixed τu = 25 ms, τw = 60 ms and varying a ∈ [10, 800]. Decay rate r as modulus of the eigenvalue λ1 and angle ϕ as argument of λ1 are shown for a = 282, marked with * and ** for EF and SE respectively. The gray half-circle denotes the stable region of r ≤ 1. e Relationship of parameter a to intrinsic frequency f (top) and decay rate r (bottom) for the same τu and τw as in panel d. Points with * and ** denote the corresponding eigenvalues from panel b. Recall the linear relationship \(f=\frac{\phi }{2\pi \Delta t}\) between angle ϕ and f of the discrete models. Horizontal gray line in bottom panel denotes stability boundary of r = 1. Values for r of continuous model (r = 0.897) and SE-adLIF (r = 0.887) are constant w.r.t. a (SE-adLIF values for r not visible due to near-perfect fit to the continuous model). f Maximum admissible frequency for stable dynamics for Euler-Forward discretization with Δt = 1 ms over different values of τu and τw, where a is set to the maximum stable value \({a}_{\max }\) (see main text and Section Stable ranges for intrinsic frequencies of EF-adLIF in “Methods”).
