Dept. of Physiology, Anatomy and Genetics, University of Oxford, Oxford, UK
Mathematical neuron models can be described as hybrid discrete/continuous systems, with the continuous part of the system being expressed as differential equations (Brette et al., 2007). The Parker-Sochacki (PS) method is a new technique for the numerical integration of differential equations that works by iteratively computing the Maclaurin series for each variable in a system of polynomial equations.
We applied the Parker-Sochacki method to two neuron models: the Izhikevich ‘simple’ model and a Hodgkin-Huxley (HH) type neuron. The Izhikevich model (Izhikevich, 2003, 2007) is a two variable, phenomenological neuron model, featuring a quadratic membrane potential and a linear recovery variable. The Izhikevich model neuron is capable of a rich dynamic repertoire; it is the simplest model capable of spiking, bursting, and being either an integrator or a resonator (Izhikevich, 2007). Since the model equations are polynomial, the PS method can be applied directly here. We show that effective adaptive error control can be achieved by locally adapting the solution order without employing adaptive stepsize control. Using this approach, we develop an adaptive-order solution algorithm for the Izhikevich model that solves for exact (floating point) spike times despite using large fixed time steps. Exact synaptic event times are also accommodated.
HH models feature equations that are not polynomial. Taking the HH model from (Brette et al., 2007) as an example, we show how to employ variable substitutions to arrive at a system solvable using the PS method. Furthermore, we demonstrate that equations featuring rational functions and compositions thereof are also solvable using the PS method. The solution algorithm features adaptive order processing and exact synaptic event times.
Benchmark simulations test our PS algorithms against the Bulirsch-Stoer (BS) and 4th-order Runge-Kutta (RK) methods. The Izhikevich network model is based on Benchmark 1 from Brette et al., (2007). It features 4000 cells (80% excitatory, 20% inhibitory) randomly (2%) connected via conductance-based synapses. Using a quantitative measure of accuracy, and taking performance as accuracy/CPU time, the PS method performs 2.6 times better than RK and 22.1 times better than BS.
The HH model benchmark simulations expose a single neuron to sequences of synaptic inputs recorded during Izhikevich model simulations. Using a similar quantitative measure of performance, the PS method performs 71.3 times better than RK and 9.6 times better than BS here.
Support: The Wellcome Trust