# Mathematical model for damped oscillations in subthreshold membrane potential produced by calcium-dependent potassium channels

## Class project for Physiological systems modeling (2009), taught by Dr. Jit Muthuswamy

We have derived a simplified model of the Traub neuron calcium dynamics. This system consists of a feedback loop between membrane voltage, voltage-gated calcium channels, and calcium-dependent potassium channels. Using both eigenvalue analysis of the system's Jacobian, and analysis of phase space trajectories, we extract information about the oscillatory behavioural regimes of the system. Our findings indicate that, for our reduced model, intracellular calcium pool decay rate is the critical parameter affecting emergence of oscillations in response to perturbation.

Phase portraits for system with two oscillatory (green and red) and two non-oscillatory (blue and magenta) values for calcium decay constant. Calcium concentration is plotted vertically, while calcium-dependent potassium (KCa) current is horizontal.

Eigenvalue analysis (left) provides estimates of the rate of exponential decay (real) and oscillation frequency (imag, imaginary component) of subthreshold calcium oscillations. Data are plotted as a function of system parameter b, which quantifies the rate of calcium clearance from the intracellular pool. Different colour plots show different system configurations (varying KCa conductance and activation kinetics). System shows oscillatory regimes only for specific values of b, the width of which depend on other system parameters. (Right) Same information obtained numerically by analysis of phase plot trajectories, in agreement with analytical result.