Fitting bifurcation structure, not voltage traces: Reduced neuron models that preserve physiological parameter dependence

In conductance-based models, spiking-induced ion concentration fluctuations can alter the excitability of single neurons. In particular, in class I models, variations in potassium concentration can induce qualitative changes in the dynamics through a codimension-2 bifurcation known as the saddle-node-loop (SNL). Investigating the implications of such effects at the level of neuronal networks will require computationally efficient single neuron models that still capture ion concentration dynamics realistically. To this end, we propose a method to derive a phenomenological model capturing the coupled extracellular potassium and voltage dynamics of a given class I conductance-based model. Rather than fitting voltage traces, we calibrate a canonical reduced model to the two-parameter bifurcation structure of the target model, with input current and a physiological parameter as coordinates. This preserves the location and type of dynamical transitions as the physiological parameter varies, allowing a single reduced model to capture neuronal dynamics across qualitatively different regimes. The resulting model is an extension of the quadratic integrate-and-fire model, in which extracellular potassium accumulation alters voltage dynamics by increasing the reset voltage. We apply our systematic reduction procedure to the Wang-Buzsáki model. Its phenomenological version exhibits quantitatively comparable dynamics and replicates the reshaping of the phase-response curve associated with the transition from SNIC to HOM spikes at elevated potassium. To illustrate the derived model’s applicability, we perform a preliminary investigation of how changes in potassium concentration influence synchronization in networks.