Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process

Abstract

Stochastic leaky integrate-and-fire models are popular due to their simplicity and statistical tractability. They have been widely applied to gain understanding of the underlying mechanisms for spike timing in neurons, and have served as building blocks for more elaborate models. Especially the Ornstein–Uhlenbeck process is popular to describe the stochastic fluctuations in the membrane potential of a neuron, but also other models like the square-root model or models with a non-linear drift are sometimes applied. Data that can be described by such models have to be stationary and thus, the simple models can only be applied over short time windows. However, experimental data show varying time constants, state dependent noise, a graded firing threshold and time-inhomogeneous input. In the present study we build a jump diffusion model that incorporates these features, and introduce a firing mechanism with a state dependent intensity. In addition, we suggest statistical methods to estimate all unknown quantities and apply these to analyze turtle motoneuron membrane potentials. Finally, simulated and real data are compared and discussed. We find that a square-root diffusion describes the data much better than an Ornstein–Uhlenbeck process with constant diffusion coefficient. Further, the membrane time constant decreases with increasing depolarization, as expected from the increase in synaptic conductance. The network activity, which the neuron is exposed to, can be reasonably estimated to be a threshold version of the nerve output from the network. Moreover, the spiking characteristics are well described by a Poisson spike train with an intensity depending exponentially on the membrane potential.

Lognormal firing rate distribution reveals prominent fluctuation–driven regime in spinal motor networks

“…The probability that a given value of $V_m$ will cause a spike was estimated as the histogram of $V_m$–instances (gray histogram, Figure 4A) divided by the total time spent at all values of $V_m$ (green histogram). This gives the empirical relationship between $V_m$ and the firing rate (Jahn et al, 2011; Vestergaard and Berg, 2015). The IO–function had a strong non–linear shape (Figure 4B).…”

Synaptic inhibition and excitation estimated via the time constant of membrane potential fluctuations

“…1C). For the leaky integrate-and-fire neuronal models with linear drift, the ACF has an exponential decay with time constant equal to (Forman and Sørensen 2008;Jahn et al 2011). For more realistic models and experimental data, the ACF will not be exactly exponential, because of other underlying processes with different time constants, e.g., ligand-gated ion channel kinetics or low-pass filter properties of the membrane.…”

Spiking neurons : interacting Hawkes processes, mean field limits and oscillations

“…However there seems to be statistical evidence that a fixed firing threshold does not exist, see e.g. Jahn et al (2011) [29]. Therefore, our model does not impose a fixed firing threshold, we simply suppose that the firing rate of each neuron f i is a Lipschitz continuous, increasing function, which is reasonable from a modeling point of view.…”

“…The biological motivation for this paper comes from the rhythmic scratch like network activity in the turtle, induced by a mechanical stimulus, and recorded and analyzed by Berg and co-workers [2][3][4]29]. Oscillations in a spinal motoneuron are initiated by the sensory input, and continue by some internal mechanisms for some time after the stimulus is terminated.…”