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An important characterization of neural spiking is the ratio of the variance to the mean of the spike counts in a set of intervals--the Fano factor. For a Poisson process, the theoretical Fano factor is exactly one. For simulated or experimental neural data, the sample Fano factor is never exactly one, but often appears close to one. In this short communication, we characterize the distribution of the Fano factor for a Poisson process, allowing us to compute probability bounds and perform hypothesis tests on the distribution of recorded neural spike counts. We show that for a Poisson process the Fano factor asymptotically follows a gamma distribution with dependence on the number of observations of spike counts, and that convergence to this asymptotic distribution is fast. The analysis provides a simple method to determine how close to 1 the computed Fano factor should be and to formally test whether the observed variability in the spiking is likely to arise in data generated by a Poisson process.

Despite the success of contemporary treatment protocols in childhood acute lymphoblastic leukaemia (ALL), relapse within the central nervous system (CNS) remains a challenge. To better understand this phenomenon, we have analysed the changes in incidence and pattern of CNS relapses in 5564 children enrolled in four successive Medical Research Council-ALL trials between 1985 and 2001. Changes in the incidence and pattern of CNS relapses were examined and the relationship with patient characteristics was assessed. The factors affecting outcome after relapse were determined. Overall, relapses declined by 49%. Decreases occurred primarily in non-CNS and combined relapses with a progressive shift towards later (> or =30 months from diagnosis) relapses (P

Perception, memory, learning, and decision making are processes carried out in the brain. The performance of such intelligent tasks is made possible by the communication of neurons through sequences of voltage pulses called spike trains. It is of great interest to have methods of extracting information from spike trains in order to learn about their relationship to behavior. In this article, we review a Bayesian approach to this problem based on state-space representations of point processes. We discuss some of the theory and we describe the way these methods are used in decoding motor cortical activity, in which the hand motion is reconstructed from neural spike trains.

Summary Anthracyclines are used to treat childhood acute lymphoblastic leukaemia (ALL) but non-randomized studies suggest that cardiotoxicity may be a problem. Individual patient data from trials in childhood ALL that randomized anthracyclines or methods of reducing cardiotoxicity were analysed by standard meta-analysis methods. Results were grouped and combined according to: addition of an anthracycline to standard therapy, type of anthracycline, mode of administration, and the use of a cardioprotectant. Data from 958 patients in 4 trials, recruiting between 1972 and 1984, showed that addition of an anthracycline reduced bone marrow relapse and, non-significantly, non-bone marrow relapse, resulting in an increased relapse-free interval. However there was a non-significant increase in induction failures, and in deaths in first remission. Event-free survival at 5 years was 56·7% with anthracycline versus 52·8% without (Odds Ratio = 0·91; 95% Confidence Interval = 0·76-1·10; P = 0·3). There were no significant differences found in other treatment comparisons. The limited data from trials did not demonstrate differences in clinically evident cardiotoxicity. Anthracyclines are effective against bone marrow relapse but have not been shown to significantly increase event free survival in childhood ALL. The evidence on type of anthracycline, method of administration or use of cardioprotectant was insufficient to be able to rule out important differences. © 2009 Blackwell Publishing Ltd.

This paper is aimed at exploring the effect of cortical brain atrophy on the currents induced by transcranial magnetic stimulation (TMS). We compared the currents induced by various TMS conditions on several different MRI derived finite element head models of brain atrophy, incorporating both decreasing cortical volume and widened sulci. The current densities induced in the cortex were dependent upon the degree and type of cortical atrophy and were altered in magnitude, location, and orientation when compared to healthy head models. Predictive models of the degree of current density attenuation as a function of the scalp-to-cortex distance were analyzed, concluding that those which ignore the electromagnetic field-tissue interactions lead to inaccurate conclusions. Ultimately, the precise site and population of neural elements stimulated by TMS in an atrophic brain cannot be predicted based on healthy head models which ignore the effects of the altered cortex on the stimulating currents. Clinical applications of TMS should be carefully considered in light of these findings.

Behavioral tests are widely used to quantify features of cognitive processing. For a large class of behavioral signals, the observed variables are non-Gaussian and dynamic; classical estimation algorithms are ill-suited to modeling such data. In this research, we propose a mathematical framework to predict a cognitive state variable related to behavioral signals, which are best modeled using a Gamma distribution. The proposed algorithm combines a Gamma Smoother and EM algorithm in the prediction process. The algorithm is applied to reaction time recorded from subjects performing a Multi-Source Interference Task (MSIT) to dynamically quantify their cognitive flexibility through the course of the experiment.

The stochastic state point process filter (SSPPF) and steepest descent point process filter (SDPPF) are adaptive filter algorithms for state estimation from point process observations that have been used to track neural receptive field plasticity and to decode the representations of biological signals in ensemble neural spiking activity. The SSPPF and SDPPF are constructed using, respectively, Gaussian and steepest descent approximations to the standard Bayes and Chapman-Kolmogorov (BCK) system of filter equations. To extend these approaches for constructing point process adaptive filters, we develop sequential Monte Carlo (SMC) approximations to the BCK equations in which the SSPPF and SDPPF serve as the proposal densities. We term the two new SMC point process filters SMC-PPFs and SMC-PPFD, respectively. We illustrate the new filter algorithms by decoding the wind stimulus magnitude from simulated neural spiking activity in the cricket cercal system. The SMC-PPFs and SMC-PPFD provide more accurate state estimates at low number of particles than a conventional bootstrap SMC filter algorithm in which the state transition probability density is the proposal density. We also use the SMC-PPFs algorithm to track the temporal evolution of a spatial receptive field of a rat hippocampal neuron recorded while the animal foraged in an open environment. Our results suggest an approach for constructing point process adaptive filters using SMC methods.

The signal-to-noise ratio (SNR), a commonly used measure of fidelity in physical systems, is defined as the ratio of the squared amplitude or variance of a signal relative to the variance of the noise. This definition is not appropriate for neural systems in which spiking activity is more accurately represented as point processes. We show that the SNR estimates a ratio of expected prediction errors and extend the standard definition to one appropriate for single neurons by representing neural spiking activity using point process generalized linear models (PP-GLM). We estimate the prediction errors using the residual deviances from the PP-GLM fits. Because the deviance is an approximate χ(2) random variable, we compute a bias-corrected SNR estimate appropriate for single-neuron analysis and use the bootstrap to assess its uncertainty. In the analyses of four systems neuroscience experiments, we show that the SNRs are -10 dB to -3 dB for guinea pig auditory cortex neurons, -18 dB to -7 dB for rat thalamic neurons, -28 dB to -14 dB for monkey hippocampal neurons, and -29 dB to -20 dB for human subthalamic neurons. The new SNR definition makes explicit in the measure commonly used for physical systems the often-quoted observation that single neurons have low SNRs. The neuron's spiking history is frequently a more informative covariate for predicting spiking propensity than the applied stimulus. Our new SNR definition extends to any GLM system in which the factors modulating the response can be expressed as separate components of a likelihood function.

Recoil polarimetry was used to extract the ratio of the proton electromagnetic form factors, μ p G E p / G M p = 0.878 ± 0.064 (stat) ± 0.012 (sys) , at Q 2 = 1.13 ( GeV / c ) 2 from the reaction H 1 ( e → , e p → ) . This was an ancillary measurement in which the proton polarization was determined as part of a larger program utilizing a stand-alone polarimeter designed to measure μ n G E n / G M n . This measurement complements previous recoil polarimetry measurements of μ p G E p / G M p made at the Thomas Jefferson National Accelerator Facility.

The fungus Beauveria bassiana is a virulent pathogen of the clover root weevil a major introduced pest of clover in New Zealand Trials to investigate establishment of fungal inoculum in pasture were conducted in the Waikato region of New Zealand Granular formulations of conidia based on biopolymer technology successfully allowed the established of fungus in pasture However it did not support survival of inoculum into the second year Conidia on rice and conidial emulsions resulted in more successful establishment These results have implications for development of a biopesticide for clover root weevil based on B bassiana

An important question in neuroscience is understanding the relationship between high-dimensional electrophysiological data and complex, dynamic behavioral data. One general strategy to address this problem is to define a low-dimensional representation of essential cognitive features describing this relationship. Here we describe a general state-space method to model and fit a low-dimensional cognitive state process that allows us to relate behavioral outcomes of various tasks to simultaneously recorded neural activity across multiple brain areas. In particular, we apply this model to data recorded in the lateral prefrontal cortex (PFC) and caudate nucleus of non-human primates as they perform learning and adaptation in a rule-switching task. First, we define a model for a cognitive state process related to learning, and estimate the progression of this learning state through the experiments. Next, we formulate a point process generalized linear model to relate the spiking activity of each PFC and caudate neuron to the stimated learning state. Then, we compute the posterior densities of the cognitive state using a recursive Bayesian decoding algorithm. We demonstrate that accurate decoding of a learning state is possible with a simple point process model of population spiking. Our analyses also allow us to compare decoding accuracy across neural populations in the PFC and caudate nucleus.

Neural receptive fields are frequently plastic: a neural response to a stimulus can change over time as a result of experience. We developed an adaptive point process filtering algorithm that allowed us to estimate the dynamics of both the spatial receptive field (spatial intensity function) and the interspike interval structure (temporal intensity function) of neural spike trains on a millisecond time scale without binning over time or space. We applied this algorithm to both simulated data and recordings of putative excitatory neurons from the CA1 region of the hippocampus and the deep layers of the entorhinal cortex (EC) of awake, behaving rats. Our simulation results demonstrate that the algorithm accurately tracks simultaneous changes in the spatial and temporal structure of the spike train. When we applied the algorithm to experimental data, we found consistent patterns of plasticity in the spatial and temporal intensity functions of both CA1 and deep EC neurons. These patterns tended to be opposite in sign, in that the spatial intensity functions of CA1 neurons showed a consistent increase over time, whereas those of deep EC neurons tended to decrease, and the temporal intensity functions of CA1 neurons showed a consistent increase only in the “theta” (75–150 msec) region, whereas those of deep EC neurons decreased in the region between 20 and 75 msec. In addition, the minority of deep EC neurons whose spatial intensity functions increased in area over time fired in a significantly more spatially specific manner than non-increasing deep EC neurons. We hypothesize that this subset of deep EC neurons may receive more direct input from CA1 and may be part of a neural circuit that transmits information about the animal's location to the neocortex.

A number of studies have examined the theta-rhythmic modulation of neuronal firing in the hippocampal circuit. For extracellular recordings, this is often done by examining spectral properties of the spike-time autocorrelogram, most significantly, for validating the presence or absence of theta modulation across species. These techniques can show significant rhythmicity for high firing rate, highly rhythmic neurons; however, they are substantially biased by several factors including the peak firing rate of the neuron, the amount of time spent in the neuron’s receptive field, and other temporal properties of the rhythmicity such as cycle-skipping. These limitations make it difficult to examine rhythmic modulation in neurons with low firing rates or when an animal has short dwell times within the firing field and difficult to compare rhythmicity under disparate experimental conditions when these factors frequently differ. Here, we describe in detail the challenges that researchers face when using these techniques and apply our findings to recent recordings from bat entorhinal grid cells, suggesting that they may have lacked enough data to examine theta rhythmicity robustly. We describe a more sensitive and statistically rigorous method using maximum likelihood estimation (MLE) of a parametric model of the lags within the autocorrelation window, which helps to alleviate some of the problems of traditional methods, which was also unable to detect rhythmicity in bat grid cells. Using large batteries of simulated data, we explored the boundaries for which the MLE technique and the theta index can detect rhythmicity. The MLE technique is less sensitive to many features of the autocorrelogram, and provides a framework for statistical testing to detect rhythmicity as well as changes in rhythmicity in individual sessions providing a substantial improvement over previous methods.

Coherence is a fundamental tool in the analysis of neuronal data and for studying multiscale interactions of single and multiunit spikes with local field potentials. However, when the coherence is used to estimate rhythmic synchrony between spiking and any other time series, the magnitude of the coherence is dependent upon the spike rate. This property is not a statistical bias, but a feature of the coherence function. This dependence confounds cross-condition comparisons of spike–field and spike–spike coherence in electrophysiological experiments.Taking inspiration from correction methods that adjust the spike rate of a recording with bootstrapping (‘thinning’), we propose a method of estimating a correction factor for the spike–field and spike–spike coherence that adjusts the coherence to account for this rate dependence.We demonstrate that the proposed rate adjustment is accurate under standard assumptions and derive distributional properties of the estimator.The reduced estimation variance serves to provide a more powerful test of cross-condition differences in spike–LFP coherence than the thinning method and does not require repeated Monte Carlo trials. We also demonstrate some of the negative consequences of failing to account for rate dependence.The proposed spike–field coherence estimator accurately adjusts the spike–field coherence with respect to rate and has well-defined distributional properties that endow the estimator with lower estimation variance than the existing adjustment method.

In this tutorial, we will introduce different models of neurons and networks of neuronal populations that are commonly used in the field of computational neuroscience. We will cover biophysical-based models of neurons and hybrid models of neural networks. We will discuss algorithms to estimate parameters of these models and describe their use in various applications including characterization of neurons in diseased networks (e.g. Parkinson's disease, epilepsy) and the effect of deep brain stimulation control. Below is a list of 4 lectures and descriptions.

The A(e,e{prime}K{sup +})YX reaction on H, D, {sup 3}He, and {sup 4}He was investigated in Hall C at CEBAF. Data were obtained for Q{sup 2} {approx} 0.35 and 0.5 GeV{sup 2} at 3.245 GeV. The missing mass spectra for both H and D are fitted with Monte-Carlo simulations incorporating peaks corresponding to {Lambda} production on the proton and {Sigma} production on both the proton and neutron. For D, the cross section ratio {Sigma}{sup 0}/{Sigma}{sup {minus}} {approx} 2, and excess yield close to the thresholds for {Lambda} and {Sigma} production can be attributed to final-state interactions that are compared to the data. The analysis of the data for the He targets is in a more preliminary state with broader quasi-free peaks resulting from the higher Fermi momenta. Evidence for bound {Lambda}-hypernuclear states is seen and other structure may be present.

A detailed investigation of the basic hyperon-nucleon interactions in nuclei is one of the aims of Experiment 91-016, approved with high priority at CEBAF, to study the electroproduction of kaons on targets of deuterium, {sup 3}He, and {sup 4}He. Inasmuch as both the electron and K{sup +} are particles that interact relatively weakly with nucleons, electroproduction of light hypernuclei provides a low-distortion method for investigating the fundamental interactions between nucleons, {Alpha}`s, and {Epsilon}`s in few-body systems. In particular, the (e,e`K{sup +}) reactions on cryogenic targets of D, {sup 3}He, and {sup 4}He will be studied at incident electron energies near 3 GeV with coincident detection of the emergent e and K{sup +} in the HMS and SOS magnetic spectrometers in Hall C. Construction of the He target, operating at {approximately}10 atm, {approximately}50 K and capable of dissipating {approximately}30 W, is expected to be complete prior to commencement of production runs in Hall C. The first data runs for E91-016, expected to begin late in FY 1995, will also be the basis for a doctoral thesis at Hampton University. In addition to providing new information on the phases of hyperon-nucleon interactions, measurements of cross sections for hypernuclear formation, and interference phenomena,more » the data may provide evidence for the presence of bound {Epsilon}`s and strange di-baryonic states that are the subject of considerable theoretical discussion.« less

A small plot trial was undertaken in the Waikato to determine the potential impact of Sitona lepidus adults on white clover seed production. Five densities of adult weevil were caged on 1 m2 plots from the time of peak adult emergence in November 1998 until harvest in late January 1999. In the laboratory, weevil adults were found to severely damage clover inflorescences when no leaves were present. Although feeding damage on leaves was highly visible in the field plots, with over 80% of the leaflets damaged at the highest density of 600 adults/m2, weevil adults were found to have no effect on flowering or seed production.

A time-zero fission fragment (FF) detector, based on the technique of low-pressure multiwire proportional chambers (LPMWPC), has been designed and constructed for the heavy hypernuclear lifetime experiment (E95-002) at Thomas Jefferson National Accelerator Facility. Its characteristics and the method of time-zero reconstruction were investigated using fission fragments from a 252 Cf spontaneous fission source. The influence of the ionization energy loss was also studied. It is shown that Heptane, Hexane, and Isobutane gases at a pressure of 1–2 Torr are all suitable for such a FF detector. As desired by experiment, a timing resolution of about 200 ps (FWHM) for a chamber size of 21×21 cm 2 was achieved.

The following works are referred to by author's surname only: H. W. Garrod, P. Papini Stati Thebais et Achilleis (Oxford, 1906); D. E. Hill, P. Papini Stati Thebaidos Libri XII [Mnemosyne Supplement 79 (Leiden, 1983)]; A. Klotz, P. Papini Stati Thebais (Leipzig, 1908; revised edn by T. C. Klinnert, 1973); R. Lesueur, Stace Thébaïde (Paris, 1990, 1991, 1994; 3 vols); J. H. Mozley, Statius (Loeb edn, London, 1928; 2 vols); J. J. Smolenaars, Statius, Thebaid VII [Mnemosyne Supplement 134 (Leiden, 1994)]; P. Venini, P. Papini Stati Thebaidos Liber XI (Firenze, 1970).

Jefferson Lab experiment E94-014 measured the excitation of the Delta (1232) resonance via the reaction p(e,e{prime}p){pi}{sup 0} at Q{sup 2} near 2.8 and 4 Gev{sup 2}. This is the highest Q{sup 2} for which exclusive resonance electroproduction has ever been observed. Decay distributions of the Delta (1232) resonance into the ppi{sup 0} final state were measured over a wide range of barycentric decay angles and energies. The goal of this experiment is to assess the transition in Q{sup 2} from the constituent quark model (CQM) to the regime where hard processes become important. At Q{sup 2} {approx} few GeV{sup 2}/c{sup 2} the ratio E{sub 1+}/M{sub 1+} depends dramatically on the theoretical description, varying from a few 10{sup -2} in the CQM limit, to about 1 in the pQCD limit. Preliminary analysis of the data shows that the ratio E{sub 1+}/M{sub 1+} remains small at Q{sup 2} up to 4 Gev{sup 2}/c{sup 2}. After first pass analysis we obtain E{sub 1+}/M{sub 1+} = -4.1 {+-} 1.2 at Q{sup 2} = 2.8 Gev{sup 2}/c{sup 2} and E{sub 1+}/M{sub 1+} = -7.9 {+-} 0.8 at Q{sup 2} = 4 Gev{sup 2}/c{sup 2} (statistical errors only).

The first measurements of the differential cross section for the d(gamma,p)n reaction up to 4.0 GeV were performed at Continuous Electron Beam Accelerator Facility (CEBAF) at Jefferson Lab. We report the cross sections at the proton center-of-mass angles of 36, 52, 69 and 89 degrees. These results are in reasonable agreement with previous measurements at lower energy. The 89 and 69 degree data show constituent-counting-rule behavior up to 4.0 GeV photon energy. The 36 and 52 degree data disagree with the counting rule behavior. The quantum chromodynamics (QCD) model of nuclear reactions involving reduced amplitudes disagrees with the present data., Comment: 5 pages (REVTeX), 1 figure (postscript)

U.S. Department of Energy, National Science Foundation, Stichting voor Fundamenteel Onderzoek der Materie (Netherlands), Korean Science and Engineering Foundation

The 1 H(e, e′K + )Λ and 1 H(e,e′K + )Σ 0 reactions were studied as a function of the squared four-momentum-transfer, Q 2 , and the virtual photon polarization, e, thus enabling the separation of the longitudinal and transverse parts of the cross section. The existence of a sizable longitudinal component for the 1 H(e,e′K + )Λ reaction seems to be indicated by these data.

The (e,e'p) reaction was studies on targets of C, Fe, and Au at momentum transfers squared Q{sup 2} of 0.6, 1.3, 1.8 and 3.3 GeV{sup 2} in a region of kinematics dominated by quasifree electron-proton scattering. Missing energy and missing momentum distributions are reasonably well described by plane wave impulse approximation calculations with Q{sup 2} and A dependent corrections that measure the attenuation of the final state protons.

The text is taken from the edition of D. E. Hill, Mnemosyne Supplement 79 (Leiden, 1983). The following works are referred to by author's surname only: H. W. Garrod, P. Papini Stati Thebais et Achilleis (Oxford, 1906); L. Håkanson, Statins Thebaid (Lund, 1973); A. Klotz, P. Papini Stati Thebais (Leipzig 1908; revised edn by T. C. Klinnert, 1973); R. Kühner, C. Stegmann, and A. Thierfelder, Ausführliche Grammatik der lateinischen Sprache (Leverkusen, 1955; 2 vols.); R. Lesueur, Stace Thébaïde (Paris, 1990, 1991, 1994; 3 vols.); J. H. Mozley, Statius (Loeb edition, London, 1928; 2 vols.); E. C. Woodcock, A New Latin Syntax (London, 1959).

We report a measurement of the neutron detection efficiency for an array of neutron detectors behind a lead-steel wall consisting of 10.16 cm thick lead bricks sandwiched between 3.18 cm steel plates. The neutron detection efficiency e was obtained from the ratio of coincidence to single-arm proton cross sections using the γ + 2 1 H → p + n reaction. Knowledge of the neutron detection efficiency was necessary to extract the cross section for the exclusive 2 H(e,′e n) 1 H reaction, which we used to obtain the elastic magnetic form factor G M n of the neutron. The measurements for three different neutron energies were carried out at the MIT-Bates Linear Accelerator Center with an incident electron beam energy of 254 MeV.

We measured the analyzing power and the efficiency of a new neutron polarimeter that was designed to measure G/sub E//sup n/, the neutron electric form factor. The polarimeter calibration was performed as experiment E377 at the Indiana University Cyclotron Facility (IUCF) with the /sup 14/C(p/spl I.oarr/,n/spl I.oarr/)/sup 14/N reaction at proton beam energies of 124.0, 164.6, and 199.7 MeV without any shielding material ahead of the polarimeter; in addition, at 164.6 MeV we measured the analyzing power with 10 cm of lead, sandwiched between 3.5-cm iron plates, ahead of the polarimeter.

One of the great ideas in data analysis is to base probability statements on large-sample approximations, which are often easy to obtain either analytically or numerically.

The notion ofhypothesis is fundamental to science. Typically it refers to an idea that might plausibly be true, and that is to be examined or “tested” with some experimental data. Sometimes, the expectation is that the data will conform to the hypothesis. In other situations, the hypothesis is introduced with the goal of refuting it.

In Section 7.2.1 we showed how the method of moments may be used to estimate the parameters of a \(Gamma(\alpha ,\beta )\) distribution, and we immediately stated that the method of maximum likelihood provides a better solution. How do we know this? In general, how should alternative methods of estimation be compared? In this chapter we lay out a series of principles that serve as guides to practice.

In Chapter 1 we said that a measurement is determined in part by a “signal” of interest, and in part by unknown factors we may call “noise.” Statistical models introduce probability distributions to describe the variation due to noise, and thereby achieve quantitative expressions of knowledge about the signal—a process we will describe more fully in Chapters 7 and 10.

Results from the prelaunch radiometric calibration of the 21-channel High Resolution Dynamics Limb Sounder (HIRDLS) flight instrument are presented. The calibration was carried out in the Department of Physics of Oxford University. Two large aperture external blackbody cavities were used to generate stable radiances at target temperatures between $\sim$ 90 and $\sim$ 320 K. These blackbodies were located, along with the HIRDLS instrument, inside a large vacuum chamber. Data were taken at three different focal-plane temperatures ( $\sim$ 61, $\sim$ 66, and $\sim$ 71 K). To complicate matters beyond the initial scope of the prelaunch calibration, a failure of some contamination close-out material (Kapton) that lined the inner fore-optics cavity occurred during launch, which made the original in-flight radiometric calibration procedure impossible. Accordingly, the radiometric conversion algorithm had to be changed, requiring more information from prelaunch calibration to be used than first envisioned. This paper discusses a variety of details, such as data-taking procedures, analysis methodology, associated error analyses, and necessary changes to the radiometric conversion algorithm needed for in-flight data processing.

Many experiments examine the effects of multiple experimental conditions. When each measured response from a subject is a single-number, the data are usually analyzed with analysis of variance (ANOVA) .

At the beginning of this book we said that we wanted to lay out the key features of what we called, “the statistical paradigm,” which consists of broadly applicable concepts that guide reasoning from data in diverse contexts. One of its foundations is the idea that data may be used to express knowledge and uncertainty about unknown values of model parameters, especially through confidence intervals.

Multiple linear regression is a powerful method of exploring relationships between a response Y and a set of potential explanatory variables \(x_1,\ldots , x_p\), but it has an obvious limitation: it assumes the predictive relationship is, on average, linear. In addition, in its standard form it assumes that the noise contributions are homogeneous and follow, roughly, a normal distribution.

The examples in previous chapters, involving experimental settings ranging from human and animal behavior, to neuroimaging, EEG and EMG, neural spike trains, and in vitro recording, have illustrated the way statistical models describe regularity and variability of neural data.

In Chapter 10 we laid out the main ideas in assessing statistical significance. First, there is a null hypothesis; second, there is a statistic that defines some deviation away from a null model; third there is a \(p\)-value to judge the rarity of the observed deviation under the null hypothesis.

Probability is a rich and beautiful subject, a discipline unto itself. Its origins were concerned primarily with games of chance, and many lectures on elementary probability theory still contain references to dice, playing cards, and coin flips. These lottery-style scenarios remain useful because they are evocative and easy to understand.