Dynamical Origin of Independent Spiking and Bursting Activity in Neural Microcircuits

The relationship between spiking and bursting dynamics is a key question in neuroscience, particularly in understanding the origins of different neural coding strategies and the mechanisms of motor command generation and neural circuit coordination. Experiments indicate that spiking and bursting dynamics can be independent. We hypothesize that different mechanisms for spike and burst generation, intrinsic neuron dynamics for spiking and a modulational network instability for bursting, are the origin of this independence. We tested the hypothesis in a detailed dynamical analysis of a minimal inhibitory neural microcircuit (motif) of three reciprocally connected Hodgkin-Huxley neurons. We reduced this high-dimensional dynamical system to a rate model and showed that both systems have identical bifurcations from tonic spiking to burst generation, which, therefore, does not depend on the details of spiking activity.

Figure 1

Rate response $x$ of the HH neuron model (1, 2, 3, 4) to dc input currents $I_{\mathrm{dc}}$ (bullets). It is fitted almost perfectly by (10) (thin line).

Figure 2

Bifurcations of the 3 neuron HH circuit (a)–(d) in comparison to the rate model (e)–(h) for symmetric reciprocal interaction of increasing strength. The three displayed variables are the synaptic activation $\mathcal{S}$ of the synapses originating from each of the three neurons, respectively. This quantity is a low-pass filtered version of the neuron’s spike train in case of the spiking neurons and thus provides a “rate description” of the spiking activity. The bifurcations are identical in both cases and even the exact critical values of $g$ are very close. Synaptic conductances (a) $g=10\mathrm{nS}$, (b) $g=30\mathrm{nS}$, (c) $g=50\mathrm{nS}$, (d) $g=60\mathrm{nS}$, (e) $g=30\mathrm{nS}$, (f) $g=40\mathrm{nS}$, (g) $g=51.4\mathrm{nS}$, (h) $g=60\mathrm{nS}$. × denotes nodes and + saddle points.

Figure 3

Bifurcations of the symmetric rate model (obtained with AUTO-07p [14]). Nodes are crosses and saddle points circles. The control parameter $g$ is color coded, and increasing $g$ moves fixed points along the arrows. The bifurcations are (for increasing $g$) (a) three nodes and three saddles appear at 1 ($g=35.1\mathrm{nS}$), (b) the middle node merges with the saddles coming from 1, becomes a saddle, and three new saddles appear that move towards 3 ($g=41.58\mathrm{nS}$), (c) the three saddles reach 3, become nodes, and give rise to a pair of saddles each ($g=46.3\mathrm{nS}$), and (d) the saddles merge with the three fixed points at 4, form three new saddles, and the nodes at 3 remain the only stable fixed points ($g=51.5\mathrm{nS}$).

Figure 4

Bifurcation of the HH model (a),(b) for increasing asymmetry of the connections compared to the same transition in the rate model (c),(d). We, again, find the same structure: The saddle points move to the corners and eventually merge with the fixed points, vanish, and give rise to a globally attracting limit cycle of bursting dynamics. (a) $g_1=60\mathrm{nS}$, $g_2=45\mathrm{nS}$, (b) $g_1=80\mathrm{nS}$, $g_2=35\mathrm{nS}$, (c) $g_1=60\mathrm{nS}$, $g_2=50\mathrm{nS}$, and (d) $g_1=80\mathrm{nS}$, $g_2=35\mathrm{nS}$.

Figure 5

(a)–(c) Time series of the membrane potentials $V_i$ of the three neurons. (a) $I_{\mathrm{stim}}=0.08\mathrm{nA}$, (b) $I_{\mathrm{stim}}=0.16\mathrm{nA}$, and (c) $I_{\mathrm{stim}}=0.22\mathrm{nA}$. The higher the intrinsic spiking rate of the neuron (determined by $I_{\mathrm{stim}}$), the closer comes the limit cycle to the heteroclinic cycle. For (c) the heteroclinic cycle is attracting and the time series of bursts continues to slow down infinitely. (d) Rate ($\mathcal{S}$ space) picture of the dynamics in (c), approaching the heteroclinic cycle.