Clustering through postinhibitory rebound in synaptically coupled neurons

Postinhibitory rebound is a nonlinear phenomenon present in a variety of nerve cells. Following a period of hyperpolarization this effect allows a neuron to fire a spike or packet of spikes before returning to rest. It is an important mechanism underlying central pattern generation for heartbeat, swimming and other motor patterns in many neuronal systems. In this paper we consider how networks of neurons, which do not intrinsically oscillate, may make use of inhibitory synaptic connections to generate large scale coherent rhythms in the form of cluster states. We distinguish between two cases (i) where the rebound mechanism is due to anode break excitation and (ii) where rebound is due to a slow T-type calcium current. In the former case we use a geometric analysis of a McKean-type model to obtain expressions for the number of clusters in terms of the speed and strength of synaptic coupling. Results are found to be in good qualitative agreement with numerical simulations of the more detailed Hodgkin-Huxley model. In the second case we consider a particular firing rate model of a neuron with a slow calcium current that admits to an exact analysis. Once again existence regions for cluster states are explicitly calculated. Both mechanisms are shown to prefer globally synchronous states for slow synapses as long as the strength of coupling is sufficiently large. With a decrease in the duration of synaptic inhibition both systems are found to break into clusters. A major difference between the two mechanisms for cluster generation is that anode break excitation can support clusters with several groups, while slow T-type calcium currents predominantly give rise to clusters of just two (antisynchronous) populations.

Figure 1

(Color online) The phase-plane for the reduced Hodgkin-Huxley model obtained by the method of equivalent potentials. The straight diagonal line is the $w$ nullcline, $w=v$, while the two “cubic'' curves are the $v$ nullclines with $I=0$ and $I=-4$. The inset shows the rebound spike that arises when the fixed point with $I=-4$ is abruptly removed, by setting $I=0$.

Figure 2

(Color online) The phase-plane for the modified McKean model, with $v$ nullclines plotted for $I=0$ and $I=-4$. Also shown is a rebound spike created by removing inhibition from the rest state with $I=-4$. $\mu =0.01$, $w\left(x\right)=4\log (1+x∕4.75)$ and other parameters as in Fig. 1.

Figure 3

(Color online) A plot of the critical curves $\alpha \left(M\right)$ as a function of $J$ (i.e., synaptic speed vs synaptic strength), defining the regions of existence for $M$-cluster states. Here we choose $w\left(x\right)=x$ and all other parameters as in Fig. 2. Note the coexistence of states.

Figure 4

(Color online) Direct numerical simulations of a network of $N=120$ Hodgkin-Huxley neurons, showing the coexistence of a 1-cluster (top) and a 2-cluster (bottom) state for $\alpha =0.1$ and $J=20$. In the top trace all of the neurons in the globally coupled network synchronize. In the bottom figure (with a different set of initial conditions) the network splits into two equally sized clusters that oscillate in antiphase.

Figure 5

(Color online) Direct numerical simulations of a network of $N=120$ Hodgkin-Huxley neurons, showing the coexistence of a 2-cluster (top) and a 3-cluster (bottom) state for $\alpha =1.25$ and $J=200$. In the top figure the network has split into two clusters that oscillate in antiphase. In the bottom figure the network has split into three equal sized groups, such that the phase difference between clusters is uniformly distributed on the circle.

Figure 6

(Color online) Response of an IFB neuron to a hyperpolarizing current step. Note that upon release from inhibition there is a post inhibitory rebound response consisting of a burst of spikes. Here $v_{\mathrm{ss}}$ denotes the steady state of the neuron.

Figure 7

(Color online) A plot of the explicit HLC solution with $\alpha =0.05$, $g_s=2.0$, and $v_s=-100$. For these parameters we find $\Delta =112.9$, ${\Delta }^{+}=26.4$, and ${\Delta }_{\theta }=11.5$.

Figure 8

(Color online) A plot of the voltage for a network of $N=100$ neurons. Bright colors denote high activity and dull colors low activity. All parameters as in Fig. 7. Note the rapid approach to a HLC from random initial data.

Figure 9

(Color online) Voltage trace for one of the neurons in Fig. 8, showing good asymptotic agreement with the analytically calculated orbit of Fig. 7.

Figure 10

(Color online) A plot of the 2-cluster state with $\alpha =0.1$. Other parameters as in Fig. 7. For these parameters we find $\Delta =99.1$, ${\Delta }^{+}=53.7$, and ${\Delta }_{\theta }=6.4$.

Figure 11

(Color online) A plot of the voltage for two neurons in a network of $N=100$ neurons. All parameters as in Fig. 10. Note the rapid approach to a 2-cluster state from random initial data, showing good asymptotic agreement with the analytically calculated orbit shown in Fig. 10. 

Figure 12

(Color online) Parameter borders encompassing regions of stable HLC and 2-cluster states in the $(\alpha ,g_s)$ plane. Other parameters as in Fig. 7. Note that the HFP is stable everywhere.