Neuronal Spike Timing Adaptation Described with a Fractional Leaky Integrate-and-Fire Model

Abstract

The voltage trace of neuronal activities can follow multiple timescale dynamics that arise from correlated membrane conductances. Such processes can result in power-law behavior in which the membrane voltage cannot be characterized with a single time constant. The emergent effect of these membrane correlations is a non-Markovian process that can be modeled with a fractional derivative. A fractional derivative is a non-local process in which the value of the variable is determined by integrating a temporal weighted voltage trace, also called the memory trace. Here we developed and analyzed a fractional leaky integrate-and-fire model in which the exponent of the fractional derivative can vary from 0 to 1, with 1 representing the normal derivative. As the exponent of the fractional derivative decreases, the weights of the voltage trace increase. Thus, the value of the voltage is increasingly correlated with the trajectory of the voltage in the past. By varying only the fractional exponent, our model can reproduce upward and downward spike adaptations found experimentally in neocortical pyramidal cells and tectal neurons in vitro. The model also produces spikes with longer first-spike latency and high inter-spike variability with power-law distribution. We further analyze spike adaptation and the responses to noisy and oscillatory input. The fractional model generates reliable spike patterns in response to noisy input. Overall, the spiking activity of the fractional leaky integrate-and-fire model deviates from the spiking activity of the Markovian model and reflects the temporal accumulated intrinsic membrane dynamics that affect the response of the neuron to external stimulation.

Emergence of Fractional Kinetics in Spiny Dendrites

“…Caputo's fractional derivative is a non-local operator and for this reason, as pointed out in [22], it could be introduced to explain emergent behaviors such as the appearance of multiple timescale dynamics and memory effects, related to the complexity of the medium. In this work we derived two possible stochastic processes, CTRW and ggBm, for inert tracer diffusion in spiny dendrites that in principle give rise to the same partial differential equation for the transmembrane potential.…”

Population density equations for stochastic processes with memory kernels

“…For example, there may be a connection with the model for adaptation posed by Teka et al [41], where a fractional derivative is introduced in the LIF model itself.…”

Incorporating spike-rate adaptation into a rate code in mathematical and biological neurons

“…9 for H(t), is a reasonable guess when a single exponential process is responsible for adaptation; however, the membrane potential of real neurons, and even relatively simple mathematical descriptions like the CS model, can exhibit dynamics with multiple timescales (La Camera et al 2006;Spain et al 1991;Ulanovsky et al 2004). Power laws are useful approximations when multiple exponential processes are at work (Anderson 2001) and have been utilized successfully in computational and experimental studies of adaptation (Drew and Abbott 2006;Lundstrom et al 2008;Teka et al 2014). In GPR2, adaptation likely occurs on multiple timescales; there is a much faster component (Ͻ0.25 s), possibly viscoelastic in origin (Swerup and Rydqvist 1996), observed in response to large square-wave stretches (Birmingham et al 1999), that we ignored when using the slowly varying stretches.…”