Calcium-activated nonspecific cation current and synaptic depression promote network-dependent burst oscillations

Abstract

Central pattern generators (CPGs) produce neural-motor rhythms that often depend on specialized cellular or synaptic properties such as pacemaker neurons or alternating phases of synaptic inhibition. Motivated by experimental evidence suggesting that activity in the mammalian respiratory CPG, the preBötzinger complex, does not require either of these components, we present and analyze a mathematical model demonstrating an unconventional mechanism of rhythm generation in which glutamatergic synapses and the short-term depression of excitatory transmission play key rhythmogenic roles. Recurrent synaptic excitation triggers postsynaptic Ca(2+)-activated nonspecific cation current (I(CAN)) to initiate a network-wide burst. Robust depolarization due to I(CAN) also causes voltage-dependent spike inactivation, which diminishes recurrent excitation and thus attenuates postsynaptic Ca(2+) accumulation. Consequently, activity-dependent outward currents-produced by Na/K ATPase pumps or other ionic mechanisms-can terminate the burst and cause a transient quiescent state in the network. The recovery of sporadic spiking activity rekindles excitatory interactions and initiates a new cycle. Because synaptic inputs gate postsynaptic burst-generating conductances, this rhythm-generating mechanism represents a new paradigm that can be dubbed a 'group pacemaker' in which the basic rhythmogenic unit encompasses a fully interdependent ensemble of synaptic and intrinsic components. This conceptual framework should be considered as an alternative to traditional models when analyzing CPGs for which mechanistic details have not yet been elucidated.

Fig. 1.

Group pacemaker models. (A and B) Uncoupled (A) and coupled (B) model neurons, showing voltage (V), Ca²⁺, and synaptic dynamics (s). Neuron 1 (N₁, top) is initially tonic (A) and Neuron 2 (N₂, bottom) is initially quiescent (A), before coupling (B). (C) Self-coupled single neuron with dynamics that resemble the 2-neuron case. Color coding applies to phase-plane analyses in D and E. (D) V-Ca²⁺ bifurcation diagram (black) with solution trajectory. Solid (dashed) black curves denote stable (unstable) structures, which are either nodes or the maximal and minimal voltages along families of periodic orbits. These include a curve of steady states at approximately −20 mV, which switches from unstable to stable in a supercritical Andronov-Hopf bifurcation (AH, half-shaded arrowhead; (Ca²⁺)_AH in text) as Ca²⁺ increases, and 3 families of periodic orbits, a stable one that emerges from the AH bifurcation, an unstable family that emerges from the first, and a second stable family of larger amplitude (in maximal and minimal voltage) that emerges from the unstable periodic orbits. Red, blue, and green portions correspond to the voltage trace in C. Voltage spikes follow the large amplitude periodic orbits. Spike attenuation begins when Ca²⁺ becomes so large that this family of periodic orbits no longer exists. Because this occurs very close to (Ca²⁺)_AH, we approximate the point where spike attenuation begins by (Ca²⁺)_AH for simplicity (see text). (E) The s-Ca²⁺ phase plane showing the curve of AH points (the half-shaded arrowhead denotes (Ca²⁺)_AH), and the Ca²⁺ nullcline (Ca null), with the solution trajectory in red, blue, and green.

Fig. 2.

Noise-induced burst oscillations in the self-coupled model. (A) Oscillations and (B) quiescence depend on E_L with no added noise. (C) The addition of membrane noise (γ = 350 pA²) allows irregular bursting for low E_L. (Inset) Back-to-back bursts in this regime. (D₁) Three-dimensional (3D) bifurcation diagram with Ca²⁺ and E_L as bifurcation parameters. The red and green surfaces consist of stable nodes and saddle points, respectively; the blue surface is formed from the maximum and minimum voltages along families of periodic orbits. (D₂) V-Ca²⁺ slice of the 3D graph in D₁ with E_L = −60.5 mV showing the lower stable node branch, the branch of saddle points (which reflect spike threshold), and the minimum voltage from the family of periodic orbits that coalesce in a SNIC bifurcation. (D₃) E_L-Ca²⁺ plane showing the curve of SNIC bifurcation points and the curve of Andronov-Hopf (AH) bifurcation points from which the periodic orbit families emanate, extracted from the 3D graph in D₁. Dark and light gray regions correspond, respectively, to quiescent and oscillatory solutions in the self-coupled model with Ca²⁺ held constant.

Fig. 3.

Self-coupled model with I_pump. (A) Voltage and Na⁺ traces (Left) with an expanded and color-coded burst (Right). (B) V-Ca²⁺ bifurcation diagrams for fixed Na⁺ values; red, blue, and green trajectories correspond to A. Stable (unstable) points are drawn with solid (dashed) lines, and periodic orbits (maximal and minimal voltages) are shown with circles. The branch near −20 mV is a curve of spiral points, featuring a change in stability at the AH bifurcation from which the periodic orbits emanate. The branches at lower voltages that appear in the Center (b) and Right (c) parts of B, consist of unstable saddles (dashed lines) and stable nodes (solid lines), which intersect at a SNIC bifurcation. (C) Expanded view near the SNIC; a, b, and c refer to Left (a), Center (b), and Right (c) in B, superimposed without trajectories. (D) The burst cycle in Na⁺-Ca²⁺ state space, together with the SNIC and AH curves. Na⁺ is plotted such that lower values on the y axis reflect lower excitability, due to higher Na⁺ evoking greater I_pump (outward) current.

Fig. 4.

Self-coupled model with different outward currents. Voltage (Left) and state-space trajectories with SNIC and AH curves (Right) are shown for M-like K⁺ current (I_M), where n_M is the gating variable in n_M-Ca²⁺ state space (A); Ca²⁺-dependent K⁺ current (I_K-Ca), where z_K-Ca is the gating variable in z_K-Ca-Ca²⁺ state space (B); and persistent Na⁺ current (I_NaP), where h_NaP is the gating variable in h_NaP-Ca²⁺ state space (C). In contrast to A and B, h_NaP values are shown increasing along the y axis (Right) so that all panels on the Right represent more net outward current in the downward direction.