Neuromathematical concepts of point process theory

Models of vestibular semicircular canal afferent neuron firing activity

Semicircular canal afferent neurons transmit information about head rotation to the brain. Mathematical models of how they do this have coevolved with concepts of how brains perceive the world. A 19th-century "camera" metaphor, in which sensory neurons project an image of the world captured by sense organs into the brain, gave way to a 20th-century view of sensory nerves as communication channels providing inputs to dynamical control systems. Now, in the 21st century, brains are being modeled as Bayesian observers who infer what is happening in the world given noisy, incomplete, and distorted sense data. The semicircular canals of the vestibular apparatus provide an experimentally accessible, low-dimensional system for developing and testing dynamical Bayesian generative models of sense data. In this review, we summarize advances in mathematical modeling of information transmission by semicircular canal afferent sensory neurons since the first such model was proposed nearly a century ago. Models of information transmission by vestibular afferent neurons may provide a foundation for developing realistic models of how brains perceive the world by inferring the causes of sense data.

The case for large-size mutations

There are no laws of physics or chemistry that forbid large mutations. Therefore, the "size" of a random mutation should fit the mathematics of a Poisson point process: The number of mutations (N), versus mutation size (MS), should obey an exponential relationship. Three examples are examined: A simple 15-mutation sequence; actual experimental data involving a sequence of 56,611 random action potentials (rather than mutations); and a synthetic sequence of 65,535 random mutations. In the latter example, with an average MS of 2.22 units, the largest MS is a 25-unit giant that would be associated with major changes.

Spontaneous activity in the statoacoustic ganglion of the chicken embryo

Statoacoustic ganglion cells in the mature bird include neurons that are responsive to sound (auditory) and those that are not (nonauditory). Those that are nonauditory have been shown to innervate an otolith organ, the macula lagena, whereas auditory neurons innervate the basilar papilla. In the present study, single-unit recordings of statoacoustic ganglion cells were made in embryonic (E19, mean = 19.2 days of incubation) and hatchling (P6-P14, mean = 8.6 days posthatch) chickens. Spontaneous activity from the two age groups was compared with developmental changes. Activity was evaluated for 47 auditory, 11 nonauditory, and 6 undefined eighth nerve neurons in embryos and 29 auditory, 26 nonauditory, and 1 undefined neurons in hatchlings. For auditory neurons, spontaneous activity displayed an irregular pattern [discharge interval coefficient of variation (CV) was >0.5, mean CV for embryos was 1.46 +/- 0.58 and for hatchlings was 1.02 +/- 0.25; means +/- SD]. Embryonic discharge rates ranged from 0.05 to 97.6 spikes per second (sp/s) for all neurons (mean 18.6 +/- 16.9 sp/s). Hatchling spontaneous rates ranged from 1.2 to 185.2 sp/s (mean 66.5 +/- 39.6 sp/s). Discharge rates were significantly higher for hatchlings (P < 0.001). Many embryonic auditory neurons displayed long silent periods between irregular bursts of neural activity, a feature not seen posthatch. All regular bursting discharge patterns were correlated with heart rate in both embryos and hatchlings. Preferred intervals were visible in the time interval histograms (TIHs) of only one embryonic neuron in contrast to 55% of the neurons in posthatch animals. Generally, the embryonic auditory TIH displayed a modified quasi-Poisson distribution. Nonauditory units generally displayed regular (CV <0.5) or irregular (CV >0.5) activity and Gaussian and modified-Gaussian TIHs. Long silent periods or bursting patterns were not a characteristic of embryonic nonauditory neurons. CV varied systematically as a function of discharge rate in nonauditory but not auditory primary afferents. Minimum spike intervals (dead time) and interval modes for auditory neurons were longer in embryos (dead time: embryos 2.88 +/- 6.85 ms; hatchlings 1.50 +/- 1.76 ms; modal intervals: embryo 10.09 +/- 22.50 ms, hatchling 3.54 +/- 3.29 ms). The results show that significant developmental changes occur in spontaneous activity between E19 and posthatch. It is likely that both presynaptic and postsynaptic changes in the neuroepithelium contribute to maturational refinements during this period of development.

Single motor unit myoelectric signal analysis with nonstationary data

The information content of the myoelectric signal (MES) is commonly revealed by statistical measures in the time or frequency domain. Empirical analyses of the MES from a single motor unit have generally assumed that features are invariant with time. Theoretical and experimental work has been done to demonstrate how nonstationary behavior in the discharge statistics of a motor neuron may affect estimates of features extracted from the motor unit's contribution to the MES. Specifically, it has been shown that nonstationary behavior can markedly influence estimates of features describing motor neuron firing behavior and consequently, the low-frequency portion of the MES power spectral density. These results may help to explain the discrepancies in the literature which report empirical models of motor neuron firing statistics.

Recurrence plots of neuronal spike trains

The recently developed qualitative method of diagnosis of dynamical systems-recurrence plots-has been applied to the analysis of dynamics of neuronal spike trains recorded from cerebellum and red nucleus of anesthetized cats. Recurrence plots revealed robust and common changes in the similarity structure of interspike interval sequences as well as significant deviations from randomness in serial ordering of intervals. Recurring episodes of alike, quasi-deterministic firing patterns suggest the spontaneous modulation of the dynamical complexity of the trajectories of observed neurons. These modulations are associated with changing dynamical properties of a neuronal spike-train-generating system. Their existence is compatible with the information processing paradigm of attractor neural networks.

Identification of bursts in spike trains

A computer algorithm to identify 'bursts' in trains of spikes is described. The algorithm works by constructing a histogram of interspike intervals, then analyzing the histogram to detect the critical interval value in the distribution that represents the break between short intervals within a burst and the longer intervals between bursts. When such a value is found, it is used as the 'threshold' to determine those intervals in the spike train that lie within a burst and those that lie between bursts and, thereby, to identify the beginning and end of each burst in the train. The validity of the bursts is evaluated with a chi-square test. The performance of the algorithm and how it can be assessed is discussed.

The effect of a random initial value in neural first-passage-time models

The effect of a random initial value is examined in several stochastic integrate-and-fire neural models with a constant threshold and a constant input. The three models considered are approximations of Stein's model, namely: (1) a leaky integrator with deterministic trajectories, (2) a Wiener process with drift, and (3) an Ornstein-Uhlenbeck process. For model 1, different distributions for the initial value lead to commonly observed interspike interval distributions. For model 2, a discrete and a uniform distribution for the initial value are examined along with some parameter estimation procedures. For model 3, with a truncated normal distribution for the initial value, the coefficient of variation is shown to be greater than 1, and as the threshold becomes large the first-passage-time distribution approaches an exponential distribution. The relationships among the models and between them and previous models are also discussed, along with the robustness of the model assumptions and methods of their verification. The effects of a random initial value are found to be most pronounced at high firing rates.

Dendritic transformations on random synaptic inputs as measured from a neuron's spike train--modeling and simulation

Extracellular spike trains recorded from central nervous system neurons reflect the random activations from a multitude of presynaptic cells making contacts mainly on the extensive dendritic trees. The dendritic potential variations are propagated towards the trigger zone where action potentials are generated. In this paper, two dendritic propagation modes are modeled: passive and quasi-active. Synaptic bombardments are modeled as being applied apically, somatically, or distributed over the dendritic tree. The resulting simulated neuronal spike trains are analyzed by point process techniques. Dendritic inputs resulted in a tendency for random bursting, interspike interval histograms with a long tail and coefficients of variation larger than one. The autocorrelation histograms reflected dynamics of the dendritic tree and they were able to discriminate between a passive or a quasi-active propagation mode and between dendritic and somatic synaptic inputs.

Analysis of the fluctuations in the interspike intervals of abducens nucleus neurons during ocular fixation in the alert cat

The behavior of abducens nucleus motoneurons and internuclear neurons was recorded during eye fixation in the alert cat. The interspike interval (ISI) values during a given fixation followed a normal distribution, with coefficients of variation for motoneurons and interneurons ranging from 4.6 to 16% and 5.7 to 21.7%, respectively. The ISI mean was linearly related to its standard deviation and to the coefficient of variation. The power spectral density of ISI fluctuations did not reveal any relevant peak. The autocorrelation function of ISI fluctuations was flat except at the origin. Results suggest the presence of a gaussian white noise superimposed on the clock-like firing of the activity of abducens neurons during ocular fixation. Some physiological implications of the results are discussed.

Markov-dependency and spectral analyses on spike-counts in mesencephalic reticular neurons during sleep and attentive states

Spontaneous activities of the mesencephalic reticular formation (MRF) neurons of head-restrained cats were recorded to investigate their dynamic properties during sleep and waking. The Markov-dependency and spectral analyses were performed on the time series of counts converted from the MRF spike-train. During slow-wave sleep (SWS), MRF neurons fired with low Markovian properties and had a similar spectral-density curve as white noise; during paradoxical sleep (PS), their firing pattern showed high Markovian properties owing to low-frequency fluctuations, with spectral densities inversely proportional to frequency (the l/f spectrum). During the attentive state of bird watching (BW), intermediate Markovian properties were observed. These results confirmed both the rest theory of SWS and the activation of the brain during PS from the viewpoint of dynamic information-processing. Furthermore, the activation of the brain during PS may be greater than in BW.