A simple neuronal model with intrinsic saturation of the firing frequency.

We present a comparison of the intrinsic saturation of firing frequency in four simple neural models: leaky integrate-and-fire model, leaky integrate-and-fire model with reversal potentials, two-point leaky integrate-and-fire model, and a two-point leaky integrate-and-fire model with reversal potentials. "Two-point" means that the equivalent circuit has two nodes (dendritic and somatic) instead of one (somatic only). The results suggest that the reversal potential increases the slope of the "firing rate vs input" curve due to a smaller effective membrane time constant, but does not necessarily induce saturation of the firing rate.

Fano Factor: A Potentially Useful Information.

The Fano factor, defined as the variance-to-mean ratio of spike counts in a time window, is often used to measure the variability of neuronal spike trains. However, despite its transparent definition, careless use of the Fano factor can easily lead to distorted or even wrong results. One of the problems is the unclear dependence of the Fano factor on the spiking rate, which is often neglected or handled insufficiently.

The Jacobi diffusion process as a neuronal model.

The Jacobi process is a stochastic diffusion characterized by a linear drift and a special form of multiplicative noise which keeps the process confined between two boundaries. One example of such a process can be obtained as the diffusion limit of the Stein's model of membrane depolarization which includes both excitatory and inhibitory reversal potentials. The reversal potentials create the two boundaries between which the process is confined.

Statistics of inverse interspike intervals: The instantaneous firing rate revisited.

The rate coding hypothesis is the oldest and still one of the most accepted and investigated scenarios in neuronal activity analyses. However, the actual neuronal firing rate, while informally understood, can be mathematically defined in several different ways. These definitions yield distinct results; even their average values may differ dramatically for the simplest neuronal models.

A single spike deteriorates synaptic conductance estimation.

We investigated the estimation accuracy of synaptic conductances by analyzing simulated voltage traces generated by a Hodgkin-Huxley type model. We show that even a single spike substantially deteriorates the estimation. We also demonstrate that two approaches, namely, negative current injection and spike removal, can ameliorate this deterioration.

Accuracy of rate coding: When shorter time window and higher spontaneous activity help.

It is widely accepted that neuronal firing rates contain a significant amount of information about the stimulus intensity. Nevertheless, theoretical studies on the coding accuracy inferred from the exact spike counting distributions are rare. We present an analysis based on the number of observed spikes assuming the stochastic perfect integrate-and-fire model with a change point, representing the stimulus onset, for which we calculate the corresponding Fisher information to investigate the accuracy of rate coding.

Presynaptic Spontaneous Activity Enhances the Accuracy of Latency Coding.

The time to the first spike after stimulus onset typically varies with the stimulation intensity. Experimental evidence suggests that neural systems use such response latency to encode information about the stimulus. We investigate the decoding accuracy of the latency code in relation to the level of noise in the form of presynaptic spontaneous activity.

The Gamma renewal process as an output of the diffusion leaky integrate-and-fire neuronal model.

Statistical properties of spike trains as well as other neurophysiological data suggest a number of mathematical models of neurons. These models range from entirely descriptive ones to those deduced from the properties of the real neurons. One of them, the diffusion leaky integrate-and-fire neuronal model, which is based on the Ornstein-Uhlenbeck (OU) stochastic process that is restricted by an absorbing barrier, can describe a wide range of neuronal activity in terms of its parameters.

Preface.

This Special Issue of Mathematical Biosciences and Engineering contains 11 selected papers presented at the Neural Coding 2014 workshop. The workshop was held in the royal city of Versailles in France, October 6-10, 2014. This was the 11th of a series of international workshops on this subject, the first held in Prague (1995), then Versailles (1997), Osaka (1999), Plymouth (2001), Aulla (2003), Marburg (2005), Montevideo (2007), Tainan (2009), Limassol (2010), and again in Prague (2012).

Coding accuracy on the psychophysical scale.

Sensory neurons are often reported to adjust their coding accuracy to the stimulus statistics. The observed match is not always perfect and the maximal accuracy does not align with the most frequent stimuli. As an alternative to a physiological explanation we show that the match critically depends on the chosen stimulus measurement scale.

Shot-noise Fano factor.

A variability measure of the times of uniform events based on a shot-noise process is proposed and studied. The measure is inspired by the Fano factor, which we generalize by considering the time-weighted influence of the events given by a shot-noise response function. The sequence of events is assumed to be an equilibrium renewal process, and based on this assumption we present formulas describing the behavior of the variability measure.

Sampling times influence the estimate of parameters in the Weibull dissolution model.

The aim is to determine how well the parameters of the Weibull model of dissolution can be estimated in dependency on the chosen times to measure the empirical data. The approach is based on the theory of Fisher information. We show that in order to obtain the best estimates the data should be collected at time instants when tablets actively dissolve or at their close proximity.

Altered intensity coding in the salicylate-overdose animal model of tinnitus.

Tinnitus is one of the leading disorders of hearing with no effective cure as its pathophysiological mechanisms remain unclear. While the sensitivity to sound is well-known to be affected, exactly how intensity coding per se is altered remains unclear. To address this issue, we used a salicylate-overdose animal model of tinnitus to measure auditory cortical evoked potentials at various stimulus levels, and analyzed on single-trial basis the response strength and its variance for the computation of the lower bound of Fisher information.

Estimation of the synaptic input firing rates and characterization of the stimulation effects in an auditory neuron.

To understand information processing in neuronal circuits, it is important to infer how a sensory stimulus impacts on the synaptic input to a neuron. An increase in neuronal firing during the stimulation results from pure excitation or from a combination of excitation and inhibition. Here, we develop a method for estimating the rates of the excitatory and inhibitory synaptic inputs from a membrane voltage trace of a neuron.

Performance breakdown in optimal stimulus decoding.

Objective: One of the primary goals of neuroscience is to understand how neurons encode and process information about their environment. The problem is often approached indirectly by examining the degree to which the neuronal response reflects the stimulus feature of interest.

Approach: In this context, the methods of signal estimation and detection theory provide the theoretical limits on the decoding accuracy with which the stimulus can be identified.

A review of the methods for neuronal response latency estimation.

Neuronal response latency is usually vaguely defined as the delay between the stimulus onset and the beginning of the response. It contains important information for the understanding of the temporal code. For this reason, the detection of the response latency has been extensively studied in the last twenty years, yielding different estimation methods.

Stein's neuronal model with pooled renewal input.

The input of Stein's model of a single neuron is usually described by using a Poisson process, which is assumed to represent the behaviour of spikes pooled from a large number of presynaptic spike trains. However, such a description of the input is not always appropriate as the variability cannot be separated from the intensity. Therefore, we create and study Stein's model with a more general input, a sum of equilibrium renewal processes.

Coding accuracy is not fully determined by the neuronal model.

It is automatically assumed that the accuracy with which a stimulus can be decoded is entirely determined by the properties of the neuronal system. We challenge this perspective by showing that the identification of pure tone intensities in an auditory nerve fiber depends on both the stochastic response model and the arbitrarily chosen stimulus units. We expose an apparently paradoxical situation in which it is impossible to decide whether loud or quiet tones are encoded more precisely.

Estimating latency from inhibitory input.

Stimulus response latency is the time period between the presentation of a stimulus and the occurrence of a change in the neural firing evoked by the stimulation. The response latency has been explored and estimation methods proposed mostly for excitatory stimuli, which means that the neuron reacts to the stimulus by an increase in the firing rate. We focus on the estimation of the response latency in the case of inhibitory stimuli.

Preface.

This Special Issue of Mathematical Biosciences and Engineering contains ten selected papers presented at the Neural Coding 2012 workshop. Neuroscience is traditionally very close to mathematics which stems from the famous theoretical work of McCulloch--Pitts and Hodgkin--Huxley in the middle of the previous century. Great progress has been made since those times and through the decades this fruitful combination of disciplines continue.

Fano factor estimation.

Fano factor is one of the most widely used measures of variability of spike trains. Its standard estimator is the ratio of sample variance to sample mean of spike counts observed in a time window and the quality of the estimator strongly depends on the length of the window. We investigate this dependence under the assumption that the spike train behaves as an equilibrium renewal process.

Responses of single neurons and neuronal ensembles in frog first- and second-order olfactory neurons.

A major challenge in sensory neuroscience is to elucidate the coding and processing of stimulus representations in successive populations of neurons. Here we recorded the spiking activity of receptor neurons (RNs) and mitral/tufted cells (MCs) in the frog olfactory epithelium and olfactory bulb respectively, in response to four odorants applied at precisely controlled concentrations. We compared how RN responses are translated in MCs.

Population coding is essential for rapid information processing in the moth antennal lobe.

We investigated how odorant information is transmitted by neurons in the moth antennal lobe (AL). The neurons were repeatedly stimulated by three different odorants and their activity was intracellularly recorded. First, the response properties of single neurons were analyzed.

Information capacity and its approximations under metabolic cost in a simple homogeneous population of neurons.

We calculate and analyze the information capacity-achieving conditions and their approximations in a simple neuronal system. The input-output properties of individual neurons are described by an empirical stimulus-response relationship and the metabolic cost of neuronal activity is taken into account. The exact (numerical) results are compared with a popular "low-noise" approximation method which employs the concepts of parameter estimation theory.

Metabolic cost of neuronal information in an empirical stimulus-response model.

The limits on maximum information that can be transferred by single neurons may help us to understand how sensory and other information is being processed in the brain. According to the efficient-coding hypothesis (Barlow, Sensory Comunication, MIT press, Cambridge, 1961), neurons are adapted to the statistical properties of the signals to which they are exposed. In this paper we employ methods of information theory to calculate, both exactly (numerically) and approximately, the ultimate limits on reliable information transmission for an empirical neuronal model.