Firing statistics in the bistable regime of neurons with homoclinic spike generation

Neuronal voltage dynamics of regularly firing neurons typically has one stable attractor: either a fixed point (like in the subthreshold regime) or a limit cycle that defines the tonic firing of action potentials (in the suprathreshold regime). In two of the three spike onset bifurcation sequences that are known to give rise to all-or-none type action potentials, however, the resting-state fixed point and limit cycle spiking can coexist in an intermediate regime, resulting in bistable dynamics. Here, noise can induce switches between the attractors, i.e., between rest and spiking, and thus increase the variability of the spike train compared to neurons with only one stable attractor. Qualitative features of the resulting spike statistics depend on the spike onset bifurcations. This paper focuses on the creation of the spiking limit cycle via the saddle-homoclinic orbit (HOM) bifurcation and derives interspike interval (ISI) densities for a conductance-based neuron model in the bistable regime. The ISI densities of bistable homoclinic neurons are found to be unimodal yet distinct from the inverse Gaussian distribution associated with the saddle-node-on-invariant-cycle bifurcation. It is demonstrated that for the HOM bifurcation the transition between rest and spiking is mainly determined along the downstroke of the action potential—a dynamical feature that is not captured by the commonly used reset neuron models. The deduced spike statistics can help to identify HOM dynamics in experimental data.

Figure 1

Bistability in homoclinic neurons. (a) Voltage trace of a homoclinic neuron (model definition in Sec. 6, with gating time constant of ${\tau }_n=0.16\mathrm{ms}$) driven with $I=4.4\mu \mathrm{A}/{\mathrm{cm}}^2$ and a noise strength of $\sigma =24\mathrm{mV}/\sqrt{\mathrm{s}}$ ($\frac{{\sigma }^2}{2}$ is the voltage diffusion constant in millivolt per second). (b) The homoclinic regime is reached by decreasing the timescale of the gating variable, ${\tau }_n$. Going from the SNIC to the homoclinic regime, the neuron passes the codimension-2 SNL bifurcation, which is common to all class-1 excitable neurons. The SNL bifurcation thus acts as a gate to the bistable, homoclinic spiking regime (shaded area). (c) Phase plots (gating variable vs voltage) of neurons in different dynamical regimes; see panel (b) for parameter-space organization. (d) Horizontal transversal of the bifurcation diagram in (b) below the SNL point. Bistability of rest and spiking leads to hysteresis in the frequency-input curve of homoclinic neurons. Within the bistable region, noise can switch between rest and spiking.

Figure 2

Depicted are left and right eigenvectors of the saddle as well as the limit cycle (solid line) and the separatrix (dashed line). (a) At the HOM bifurcation, the homoclinic orbit overlaps with the separatrix. (b) For higher input amplitudes, the separatrix is shifted away from the limit cycle.

Figure 3

Equivalent double-well potential. (a) At the saddle ($\circ$), ${\mathit{r}}_2$ is tangent to the stable manifold. Its orthogonal complement is ${\mathit{l}}_1$, onto which the node ($•$) and the minimum distance of the limit cycle ($d_{\mathrm{LC}}$) are projected. (b) Simulated particle density in the projected coordinate $y$.

Figure 4

Comparison of theoretical prediction (lines) and numerical simulations (markers) for different gating time constants ${\tau }_{\mathrm{n}}=0.155$, 0.16, and 0.165ms. (a) Distance between limit cycle and separatrix, $d_{\mathrm{LC}}$, vs input current as given by Eq. (11). (b) Mixing factor $\varpi$ as a function of $d_{\mathrm{LC}}$. (c) $1-\varpi$ vs input current.

Figure 5

Near spike onset, the analytical escape rate (solid line) fits the probability density of the escape duration from the fixed point to the separatrix (${\tau }_{\mathrm{n}}=0.165\mathrm{ms}$).

Figure 6

Interspike interval density for a numerical simulation with ${\tau }_{\mathrm{n}}=0.16\mathrm{ms}$, driven with $I=4.4\mu \mathrm{A}/{\mathrm{cm}}^2$ plus current noise with $\sigma =0.8\sqrt{\mathrm{ms}}\mu \mathrm{A}/{\mathrm{cm}}^2$. Mean ISI is 4.53 ms; coefficient of variation is 1.69.

Figure 7

Burst-length statistics fitted using a geometric distribution and the splitting probability from Eq. (10).

Figure 8

(a) Phase portrait and voltage trace of a stochastically bursting subcritical Hopf neuron are shown in gray. Stable and unstable limit cycles are shown as solid and dashed black lines. The bistability region is found for dc input parameters between a fold of limit cycles and a subcritical Hopf bifurcation. The separatrix is an unstable limit cycle. (b) (A)–(D) show phase portraits before, after, and during the emergence of the unstable limit cycle. Multimodal ISI histogram (c) for noise parameters derived from the analytically approximated unstable limit cycle in (a) and shown as the dotted ellipse in (a). The parameter of the neuron model can be found in [29], Fig. 6.16.

Figure 9

Location of the limit cycle. (a) For inputs above spike onset, the limit cycle position remains relatively stable, while saddle (open circle), and thus separatrix, move. (b) The downstroke of the limit cycle aligns with ${\mathit{r}}_2$ (green dotted line). Nullclines are shown for voltage (dark gray) and gating variable (light gray). (c) During the spike (violet voltage trace), the ionic current $I_{\mathrm{ion}}$ (red) is orders of magnitude larger than the input current $I$ (green horizontal line). The shaded area marks the area in which $I_{\mathrm{ion}}<10I$.