Coexistence of tonic firing and bursting in cortical neurons

Flavio Fröhlich and Maxim Bazhenov

Phys. Rev. E 74, 031922 – Published 28 September 2006

Abstract

Sustained neuronal activity can be broadly classified as either tonic firing or bursting. These two major patterns of neuronal oscillations are state dependent and may coexist. The dynamics and intracellular mechanisms of transitions between tonic firing and bursting in cortical networks remain poorly understood. Here we describe a detailed two-compartment conductance-based cortical neuron model which exhibits bistability with hysteresis between tonic firing and bursting for elevated extracellular potassium concentration. The study explains the ionic and dynamical mechanisms of burst generation and reveals the conditions underlying coexistence of two different oscillatory modes as a function of neuronal excitability.

Received 4 April 2006

DOI:https://doi.org/10.1103/PhysRevE.74.031922

Authors & Affiliations

Flavio Fröhlich^1,2 and Maxim Bazhenov¹

¹Salk Institute, 10010 North Torrey Pines Rd., La Jolla, California 92037, USA
²Division of Biology, University of California, San Diego, La Jolla, California 92093, USA

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Issue

Vol. 74, Iss. 3 — September 2006

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Images

Figure 1
Top panels: Representative single unit activity for ${[K^{+}]}_{o} = 2.5 mM$ (top) and ${[K^{+}]}_{o} = 6.5 mM$ (bottom) in hippocampal region CA3. Ten traces are aligned by the first spike in a sequence. In high ${[K^{+}]}_{o}$ solution, the neurons fired bursts with several spikes. Scale bar: $5 ms$ . Bottom panel: Probability distribution of instantaneous frequencies (calculated as inverse of interspike intervals, $N = 204$ ) for two cells for each ${[K^{+}]}_{o}$ concentration. Cells 1 and 2 $({[K^{+}]}_{o} = 6.5 mM)$ exhibit nonzero probabilities for instantaneous frequencies higher than $130 Hz$ , corresponding to spiking during bursts. Cells 3 and 4 $({[K^{+}]}_{o} = 2.5 mM)$ never show instantaneous frequencies exceeding $4.8 Hz$ (that corresponds to interspike intervals $> 208 ms$ ).Reuse & Permissions
Figure 2
Poincaré cross section for gradually increasing (top) and decreasing (bottom) ${[K^{+}]}_{o}$ . Tonic firing corresponds to a single point, spike doublets to two points, and bursting to a series of points in the Poincaré cross section for a given value of ${[K^{+}]}_{o}$ . Bistability between tonic firing and bursting for ${[K^{+}]}_{o} ∊ {5.75, 6.4} mM$ .Reuse & Permissions
Figure 3
(Color online) Bifurcation diagram for ${[K^{+}]}_{o} = 5.9 mM$ . Stable fixed points $P_{s}^{up}$ and $P_{s}^{down}$ (thick dashed line) are connected by the branch of unstable fixed points $P_{u}$ (thin dashed line). Solid lines indicate stable (thick) and unstable (thin) limit cycles. $O_{1}$ , Andronov-Hopf; $O_{2}$ and $O_{3}$ , fold; $O_{4}$ , saddle homoclinic orbit bifurcation points. Insets show bursting and tonic spiking patterns in the complete system with freely running $g_{K Ca}$ . Middle and bottom panels, enlarged region of interest. $O_{5}$ , Neimark-Sacker and $O_{6}$ , period doubling bifurcation points. $L_{s}$ indicates stable limit cycles. Projection of the phase trajectory for the complete system during bursting mode (solid red line, middle panel) and tonic firing (solid vertical blue line, bottom panel).Reuse & Permissions
Figure 4
(Color online) Three-dimensional bifurcation diagram for ${[K^{+}]}_{o} = 5.5 mM$ . The $Z$ axis shows the activation variable of the high-threshold ${Ca}^{2 +}$ current, $m_{I Ca}$ . A small perturbation $(g_{K Ca} = 0.0045 mS ∕ {cm}^{2})$ left from $O_{3}$ $(g_{K Ca} = 0.005 mS ∕ {cm}^{2})$ leads to convergence to the limit cycle (blue line starting close to $O_{3}$ ). A larger perturbation $(g_{K Ca} = 0.001 mS ∕ {cm}^{2})$ triggers convergence to the stable fixed point $P_{s}^{up}$ (red line starting further away from $O_{3}$ ). Left-hand insets show time courses of convergence to the stable fixed point (top, red) and limit cycle (bottom, blue). Right-hand inset shows the sequence of period doubling bifurcations in Poincaré cross section, $m_{I Ca} = 0.4$ .Reuse & Permissions
Figure 5
(Color online) Bifurcation diagram for ${[K^{+}]}_{o} = 9.0 mM$ . Projected trajectory of full system (thin solid line, red) shows transient oscillation at the end of the burst before switching to $P_{s}^{down}$ . Top inset, supercritical Andronov-Hopf bifurcation at $O_{1}$ . Bottom inset, membrane (soma, $V_{S}$ ) voltage time-course during burst.Reuse & Permissions

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