Ion concentration dynamics as a mechanism for neuronal bursting

Abstract

We describe a simple conductance-based model neuron that includes intra- and extracellular ion concentration dynamics and show that this model exhibits periodic bursting. The bursting arises as the fast-spiking behavior of the neuron is modulated by the slow oscillatory behavior in the ion concentration variables and vice versa. By separating these time scales and studying the bifurcation structure of the neuron, we catalog several qualitatively different bursting profiles that are strikingly similar to those seen in experimental preparations. Our work suggests that ion concentration dynamics may play an important role in modulating neuronal excitability in real biological systems.

Keywords: Burst; Epilepsy; Ion concentration; Neuron; Potassium; Seizure; Sodium.

Fig. 1

Bifurcation diagrams describing the neuron’s asymptotic behavior with fixed ion concentrations. In a [Na]_i = 10 mM. The detailed structure in the upper part of b is shown in Fig. 5

Fig. 2

Asymptotic behavior of the ion concentrations as k_bath (in millimolar) is varied. The curve on the left denotes stable equilibria and corresponds to k_bath ranging from 4.0 to 7.615. Projections of limit cycles (loops) are shown for k_bath = 7.62, 8.0, 8.5, and 8.95 mM; the thickness of the loops on the right reflect small, fast ion concentration fluctuations due to spiking behavior. The curve on the upper right, for k_bath ranging from 8.8 to 15.0, denotes small-amplitude loops that correspond to tonic spiking (see inset)

Fig. 3

Loops A–D represent the time evolution of the ion concentrations as the system exhibits limit cycle behavior. The loops are traversed in a counterclockwise manner. The dashed and solid lines are the SNIC and Hopf bifurcation curves, respectively

Fig. 4

Four qualitatively different bursting patterns corresponding to the four loops shown in Fig. 3. In the lower panels, solid curves represent [K]_o (left vertical axes) and dashed curves represent [Na]_i (right vertical axes). The event in d recurs with a period of approximately 16.5 s, but for clarity, only a portion of one such event is shown

Fig. 5

Magnification of the upper portion of Fig. 1b. D and E are portions of ion concentration limit cycles as described in the text. SN saddle-node, BT Bogdanov–Takens, HC homoclinic, HB Hopf, SL saddle-node loop, SNIC saddle-node infinite period

Fig. 6

Diagrams clarifying the burst termination behavior corresponding to ion concentration loops D and E of Fig. 5. Insets show complete burst time traces of the membrane voltage. In d, [Na]_i = 37.2 mM; in e, [Na]_i = 36.6 mM