Phase-locked cluster oscillations in periodically forced integrate-and-fire-or-burst neuronal populations

The minimal integrate-and-fire-or-burst neuron model succinctly describes both tonic firing and postinhibitory rebound bursting of thalamocortical cells in the sensory relay. Networks of integrate-and-fire-or-burst (IFB) neurons with slow inhibitory synaptic interactions have been shown to support stable rhythmic states, including globally synchronous and cluster oscillations, in which network-mediated inhibition cyclically generates bursting in coherent subgroups of neurons. In this paper, we introduce a reduced IFB neuronal population model to study synchronization of inhibition-mediated oscillatory bursting states to periodic excitatory input. Using numeric methods, we demonstrate the existence and stability of 1:1 phase-locked bursting oscillations in the sinusoidally forced IFB neuronal population model. Phase locking is shown to arise when periodic excitation is sufficient to pace the onset of bursting in an IFB cluster without counteracting the inhibitory interactions necessary for burst generation. Phase-locked bursting states are thus found to destabilize when periodic excitation increases in strength or frequency. Further study of the IFB neuronal population model with pulse-like periodic excitatory input illustrates that this synchronization mechanism generalizes to a broad range of n:m phase-locked bursting states across both globally synchronous and clustered oscillatory regimes.

Figure 1

The IFB model neuron response to a transient hyperpolarizing input current $I$ marked by a pale gray line. The values of the voltage threshold ($V_{\theta }$), reset ($V_{\text{reset}}$), leak reversal potential ($V_L$), and hyperpolarization level at which the low-threshold conductance $h$ engages ($V_h$) are marked with black dashed, dot-dash, dotted, and gray dashed lines, respectively. When the membrane voltage $V$ (black solid line) is hyperpolarized, the low threshold conductance variable $h$ (midgray line) increases. When the hyperpolarizing input is removed, the membrane voltage rebounds, discharging a series of spikes and $h$ returns to zero.

Figure 2

Globally synchronous and two-cluster oscillations in the IFB neuronal population model. Top: Homogeneous limit cycle solution (HLC). The upper panel shows the synchronous oscillation of the membrane potential for each cluster, $V_{1,2}$ (black line), and the cluster low-threshold conductances, $h_{1,2}$ (gray line). The lower panel shows the global population synaptic conductance $u$. Note that the cluster membrane voltage variable $V_k$ passes beyond the firing threshold at $V_{\theta }=-35$ mV. The HLC solution periodic orbit in $V$-$h$ space is shown on the right. Bottom: Antiphase two-cluster solution (two-cluster). Membrane potential and low-threshold conductance for each of the clusters oscillate in antiphase with each other. Cluster $k=1$ is shown in solid lines and $k=2$ in dotted lines. Each cluster follows the shape of the periodic orbit in $V$-$h$ shown on the right.

Figure 3

Numeric continuation of the stable periodic orbits in the IFB neuronal population model. Top left: Continuation of HLC periodic solutions with respect to $\alpha$. Stable periodic orbits are marked with solid circles and unstable periodic orbits with unfilled circles. The limit point of cycles (LPC) is also marked. The corresponding period of the stable periodic orbits is shown in the lower panel. Top right: Continuation with respect to $g_{\text{syn}}$. Stable and unstable periodic orbits are in solid and unfilled circles, and the period of the stable orbits is shown in the lower panel. Bottom: Regions of existence for the stable HLC and two-cluster periodic solutions. Two-parameter continuation of the LPC for each of the HLC and two-cluster-type orbits are marked by solid lines. Shaded areas indicate the parameter region in which the respective stable solutions exist.

Figure 4

Example of a stable 1:1 phase-locked HLC-type solution to the sinusoidally forced IFB neuronal population model. Top: HLC-type periodic solution with bursting membrane potential oscillations phase locked to the sinusoidal input. $V_{1,2}$ is marked with a solid black line, and the low-threshold conductance $h_{1,2}$ of the synchronous clusters is shown in midgray. The middle panel shows the population synaptic conductance $u$. The lower panel shows the sinusoidal conductance $u_{\text{ext}}$ in a dotted line. Bottom: Stable HLC-type periodic orbit in $V$-$h$-$u_{\text{ext}}$ space.

Figure 5

Numeric continuation of the stable 1:1 phase-locked HLC-type periodic solution. Top: Continuation with respect to $g_{\text{ext}}$ with fixed $f=15$ Hz and $\alpha =0.05$. Stable periodic orbits are marked with solid circles and unstable orbits with unfilled circles. Limit points are also marked, demarcating the isolated region of stable periodic orbits. Bottom: Two-parameter continuation of the left and right limit points of the stable 1:1 HLC-type periodic orbits with respect to $f$ and $\alpha$ for fixed $g_{\text{ext}}=0.01$. The value of $\alpha$ corresponding to the upper panel is shown by a dashed line. Solid lines mark the limit points of the stable periodic orbits, with the shaded area indicating the parameter region in which the stable 1:1 phase-locked HLC-type solutions exist.

Figure 6

Two parameter continuation of limit points of the 1:1 phase-locked HLC-type periodic solution with respect to $f$ and $g_{\text{ext}}$ and fixed $\alpha =0.05$. Center: Limit points of stable 1:1 phase-locked HLC-type periodic orbits with the shaded area indicating the region of existence. Parameter values for example solutions are marked with filled circles. Bottom left: 1:1 phase-locked HLC-type periodic solution, with $f=15$ Hz and $g_{\text{ext}}=0.01$. Average cluster membrane potential is in the top panel, low-threshold conductance is in the middle panel, and the sinusoidal input is in the lower panel. Time points at which $V_{1,2}$ crosses the threshold $V_{\theta }$ are marked with horizontal lines. Top left: HLC-type bursting solution with $f=10$ Hz and $g_{\text{ext}}=0.01$. This solution is not periodic. Bottom right: HLC-type bursting solution with $f=18$ Hz and $g_{\text{ext}}=0.005$. This solution is not periodic. Top right: HFP-type periodic solution with $f=18$ Hz and $g_{\text{ext}}=0.016$. Membrane potential oscillations are subthreshold, and the low-threshold conductance is not engaged.

Figure 7

Examples of n:m phase-locked bursting oscillations in the IFB neuronal population model with pulse-like input. Top: The IFB population displays two HLC-type population bursts to every three cycles of the input for $f=30$ Hz and $g_{\text{ext}}=0.02$. The top panel shows the membrane potential in a solid black line and the low-threshold conductance in a midgray line. The middle panel shows the global synaptic conductance, and the bottom panel shows the fast alpha synaptic conductance forced by the periodic input. Horizontal lines mark the input cycles, with solid lines indicating the cycles that elicit time-locked population bursts and dashed lines the remaining input cycles. Bottom: The IFB population displays two antiphase cluster bursts to every one cycle of the pulse-like input for $f=12$ Hz and $g_{\text{ext}}=0.02$. This state corresponds to a single periodic orbit in the full system for every input cycle, with one burst from each cluster per orbit. Trajectories for clusters $k=1$ and 2 are shown in solid and dotted lines, respectively.

Figure 8

Regions in $f$-$g_{\text{ext}}$ parameter space for which n:m phase-locked oscillatory states exist in the IFB neuronal population model with pulse-like input. Top: Phase-locked oscillatory bursting states of HLC- and two-cluster type, on the left and right, respectively. Shaded areas show the existence of phase-locked solutions, coded by the ratio of population bursts to input cycles in the periodic solution. Midgray regions indicate the IFB population supports bursting oscillations without phase locking to the input. White areas indicate parameter sets for which no oscillating solution was found, and the IFB population displayed HFP-type subthreshold oscillations with $h_{1,2}=0$. Bottom: Ratio of the population bursts to input cycles for the HLC-type and two-cluster-type periodic solutions showing overlap of different n:m phase-locked regimes with different input frequency $f$ and fixed $g_{\text{ext}}=0.015$.