Abstract
Neurons use sequences of action potentials (spikes) to convey information across neuronal networks. In neurophysiology experiments, information about external stimuli or behavioral tasks has been frequently characterized in term of neuronal firing rate. The firing rate is conventionally estimated by averaging spiking responses across multiple similar experiments (or trials). However, there exist a number of applications in neuroscience research that require firing rate to be estimated on a single trial basis. Estimating firing rate from a single trial is a challenging problem and current state-of-the-art methods do not perform well. To address this issue, we develop a new method for estimating firing rate based on a kernel smoothing technique that considers the bandwidth as a random variable with prior distribution that is adaptively updated under an empirical Bayesian framework. By carefully selecting the prior distribution together with Gaussian kernel function, an analytical expression can be achieved for the kernel bandwidth. We refer to the proposed method as Bayesian Adaptive Kernel Smoother (BAKS). We evaluate the performance of BAKS using synthetic spike train data generated by biologically plausible models: inhomogeneous Gamma (IG) and inhomogeneous inverse Gaussian (IIG). We also apply BAKS to real spike train data from non-human primate (NHP) motor and visual cortex. We benchmark the proposed method against established and previously reported methods. These include: optimized kernel smoother (OKS), variable kernel smoother (VKS), local polynomial fit (Locfit), and Bayesian adaptive regression splines (BARS). Results using both synthetic and real data demonstrate that the proposed method achieves better performance compared to competing methods. This suggests that the proposed method could be useful for understanding the encoding mechanism of neurons in cognitive-related tasks. The proposed method could also potentially improve the performance of brain-machine interface (BMI) decoder that relies on estimated firing rate as the input.
Highlights
In neural systems, signaling and interneuronal communication can be observed through the characteristic of action potentials
The spike trains were generated from inhomogeneous Gamma (IG) and inhomogeneous inverse Gaussian (IIG) models with different underlying rate functions that represent non-stationary processes usually encountered in empirical datasets
The results show the superiority of Bayesian Adaptive Kernel Smoother (BAKS) to other methods in all 18 scenarios (2 models, 3 rate functions, 3 variations of γ values) as depicted in S2 Fig. These results demonstrate the flexibility of BAKS in estimating the underlying rates under various interspike intervals (ISIs) characteristics of spike train models
Summary
In neural systems, signaling and interneuronal communication can be observed through the characteristic of action potentials (or ‘spikes’). The rate coding represents the information by the rate or frequency at which a neuron “fires” spikes, known as “firing rate”, and has been the most commonly used scheme to characterize the neuronal or network responses to external stimuli or behavioral tasks [1, 2]. The firing rate is typically estimated in offline analysis by averaging spiking responses across multiple repeated experiments known as trials. In practice spiking responses may differ considerably even though the trial setting remains approximately the same. This is partly due to the inherent stochastic nature of neurons and the difference of cognitive states during the trials [3, 4]. It is essential to be able to accurately estimate firing rate based on single trials
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