Noise, transient dynamics, and the generation of realistic interspike interval variation in square-wave burster neurons

First return maps of interspike intervals for biological neurons that generate repetitive bursts of impulses can display stereotyped structures (neuronal signatures). Such structures have been linked to the possibility of multicoding and multifunctionality in neural networks that produce and control rhythmical motor patterns. In some cases, isolating the neurons from their synaptic network reveals irregular, complex signatures that have been regarded as evidence of intrinsic, chaotic behavior. We show that incorporation of dynamical noise into minimal neuron models of square-wave bursting (either conductance-based or abstract) produces signatures akin to those observed in biological examples, without the need for fine tuning of parameters or ad hoc constructions for inducing chaotic activity. The form of the stochastic term is not strongly constrained and can approximate several possible sources of noise, e.g., random channel gating or synaptic bombardment. The cornerstone of this signature generation mechanism is the rich, transient, but deterministic dynamics inherent in the square-wave (saddle-node and homoclinic) mode of neuronal bursting. We show that noise causes the dynamics to populate a complex transient scaffolding or skeleton in state space, even for models that (without added noise) generate only periodic activity (whether in bursting or tonic spiking mode).

Figure 1

(a) Phase portrait for the noiseless neuronal model Eq. (1). A noiseless bursting trajectory is depicted in gray. The thin black curves correspond to the fast subsystem equilibrium branch $\mathcal{E}$, where solid (dashed) lines denote stable (unstable) states—saddles, in this case. The thick black “tube” depicts the envelope of the spiking manifold $\mathcal{L}$. The long dashed red (gray) curve is the $m_{\mathrm{KM}}$ nullcline: trajectories below (above) it move towards smaller (larger) $m_{\mathrm{KM}}$ values. Labelled points indicate bifurcations in the fast subsystem: SN: saddle-node and S-H: saddle-homoclinic orbit. The green (dashed black over light gray) curve defines the saddle middle branch $\mathcal{S}$, starting at the saddle node and ending at the homoclinic bifurcation. (b) Full-model trajectory (blue [dark gray] and light gray curves) emanating from initial condition (star) in the unstable manifold of a saddle in the saddle branch $\mathcal{S}$. The Poincaré surface of section $\Sigma$ is depicted in dotted beige (gray). Filled circles represent $\Sigma$ crossings, which are used in the reduced model analysis.

Figure 2

(a), (b) First return ISI maps for bursting activity of isolated LP and (c), (d) PD neurons from the STG of the lobster Panulirus interruptus. Colors (shades of gray) in panels (b) and (d) indicate successive ISI pairs at the start of bursts. ISIs have been normalized so that the maximum ISI is unity.

Figure 3

One-dimensional reduction of the dynamics. Panel (a) shows discrete dynamics $f_{1d}\left(m_{\mathrm{KM}}\right)$ of the slow variable. The highly negative derivative section close to the fixed point $m^{*}$ is of paramount importance in the suppression or addition of burstlets at the end of a burst, leading to complex ISI signatures. (b) The observable $\mathcal{T}\left\{m_{\mathrm{KM}}\right\}$ coupled to the dynamics in A, that encodes the (full model) time elapsed in a $f_{1d}\left(m_{\mathrm{KM}}\right)$ mapping. (c) The ISI signature structure, defined in terms of the observable $\mathcal{T}\{.\}$. The thin lines connecting calculated points were added to guide the eye.

Figure 4

Burstlets associated with canard trajectories. (a) The red (dark gray) part of the trajectory indicates the bursting “main sequence,” with the flux evolving through the spiking manifold $\mathcal{L}$. This particular trajectory does not fall directly to the stable part of the equilibria manifold (black curve) after $\mathcal{L}$ disappears in the homoclinic bifurcation. Instead, it glides along the unstable (saddle) branch of $\mathcal{E}$ (dashed black curve) for some time (blue [light gray] part of trajectory), until being reinjected into $\mathcal{L}$, thus generating a burstlet. The bottom traces represent the time course of the burstlet trajectory, superimposed with a regular (long trace in lighter gray) one. This trajectory was obtained by integrating the deterministic $\mathrm{Nap}\text{−}\mathrm{Kd}\text{−}\mathrm{KM}$ model for a parameter set close to a spike adding transition, so that a long “burstlet” could be easily obtained for the sake of illustration. (b)–(d) Burstlets in the stochastic $\mathrm{Nap}\text{−}\mathrm{Kd}\text{−}\mathrm{KM}$ model, a biological PD neuron and the stochastic Hindmarsh–Rose model, respectively.

Figure 5

Projection into the $m_{\mathrm{KM}}V$ plane of the flabellate structure $\mathcal{W}$ generated inside the Poincaré section $\Sigma$. The blue (long dark gray, at the top) curve depicts $\Sigma$ crossings for initial conditions close to the unstable manifolds of saddles in the $\mathcal{S}$ branch. The black circle at the end of the blue (dark gray, top) curve is visited every time a complete interburst hyperpolarization takes place, as happens to be the case for the nine-spike periodic orbit of the unperturbed model. The other colored (shades of gray, evolving counterclockwise from the long dark gray curve at the top) curves are $m_{\mathrm{KM}}V$ projections of the iterates of states in the blue (dark gray, top) curve under the Poincaré map associated with section $\Sigma$. Their (left, black) endpoints are the iterates of the left endpoint of the blue (dark gray, top) curve.

Figure 6

(a), (b) Signature skeleton (colored [shades of gray] filled circles): $\left(\mathcal{T}\right\{P\},\mathcal{T}\{f\left(P\right)\left\}\right)$ pairs for points $P$ in the flabellate structure $\mathcal{W}$ of Fig. 5. The generated “infrastructure” is only accessible by the system through the addition of noise, given the strongly attractive character of the periodic bursting orbit. The unfilled gray circles correspond to a signature generated by simulating the stochastic model. Panel (b) depicts the cluster generation mechanism for the first ISIs in a burst, along the deterministic scaffolding (colored [shades of gray] curves emerging from clusters centered at the periodic orbit for the noiseless case).

Figure 7

Isochrons of return time to $\Sigma$. Long times (blank areas) have been discarded to increase readability. The black curve is the intersection of the surface of section $\Sigma$ with the fast subsystem limit cycle manifold $\mathcal{L}$, corresponding to intraburst spiking. Red (clustered dark gray) circles correspond to $\Sigma$ crossings of an integrated trajectory of the full noisy model [Eqs. (1) and (2)]. Noise tends to spread the crossing points across isochrons, so that spikes of different positions in a burst give rise to similar ISIs—generating “kinks” in the signature. Grayscale: starting from the 0.75 ms isochron on the bottom left, return times grow monotonically following isochrons from left to right towards the rightmost isochron (black, 1.25 $\mathrm{ms}$ return time).

Figure 8

ISI signature generated by the model of a square-wave burster neuron with channel noise [Eqs. (1) and (2)]. Scaling of plots and point colors as in Fig. 2.

Figure 9

(a) Signature generated by the pre-Bötzinger neuron model with conductance noise. (b) Signature generated by the Hindmarsh–Rose neuron model with current noise. Conventions are as in Figs. 2 and 8.