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1.  Predicting the Responses of Repetitively Firing Neurons to Current Noise 
PLoS Computational Biology  2014;10(5):e1003612.
We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5–2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10–100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability. It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically.
Author Summary
Most neurons receive thousands of synaptic inputs per second. Each of these may be individually weak but collectively they shape the temporal pattern of firing by the postsynaptic neuron. If the postsynaptic neuron fires repetitively, its synaptic inputs need not directly trigger action potentials, but may instead control the timing of action potentials that would occur anyway. The phase resetting curve encapsulates the influence of an input on the timing of the next action potential, depending on its time of arrival. We measured the phase resetting curves of neurons in the subthalamic nucleus and used them to accurately predict the timing of action potentials in a phase model subjected to complex input patterns. A simple approximation to the phase model accurately predicted the changes in firing pattern evoked by dense patterns of noise pulses varying in amplitude and pulse duration, and by changes in firing rate. We also showed that the phase resetting curve changes systematically with changes in total neuron conductance, and doing so predicts corresponding changes in firing pattern. Our results indicate that the phase model may accurately represent the temporal integration of complex patterns of input to repetitively firing neurons.
doi:10.1371/journal.pcbi.1003612
PMCID: PMC4014400  PMID: 24809636
2.  Cellularly-Driven Differences in Network Synchronization Propensity Are Differentially Modulated by Firing Frequency 
PLoS Computational Biology  2011;7(5):e1002062.
Spatiotemporal pattern formation in neuronal networks depends on the interplay between cellular and network synchronization properties. The neuronal phase response curve (PRC) is an experimentally obtainable measure that characterizes the cellular response to small perturbations, and can serve as an indicator of cellular propensity for synchronization. Two broad classes of PRCs have been identified for neurons: Type I, in which small excitatory perturbations induce only advances in firing, and Type II, in which small excitatory perturbations can induce both advances and delays in firing. Interestingly, neuronal PRCs are usually attenuated with increased spiking frequency, and Type II PRCs typically exhibit a greater attenuation of the phase delay region than of the phase advance region. We found that this phenomenon arises from an interplay between the time constants of active ionic currents and the interspike interval. As a result, excitatory networks consisting of neurons with Type I PRCs responded very differently to frequency modulation compared to excitatory networks composed of neurons with Type II PRCs. Specifically, increased frequency induced a sharp decrease in synchrony of networks of Type II neurons, while frequency increases only minimally affected synchrony in networks of Type I neurons. These results are demonstrated in networks in which both types of neurons were modeled generically with the Morris-Lecar model, as well as in networks consisting of Hodgkin-Huxley-based model cortical pyramidal cells in which simulated effects of acetylcholine changed PRC type. These results are robust to different network structures, synaptic strengths and modes of driving neuronal activity, and they indicate that Type I and Type II excitatory networks may display two distinct modes of processing information.
Author Summary
Synchronization of the firing of neurons in the brain is related to many cognitive functions, such as recognizing faces, discriminating odors, and coordinating movement. It is therefore important to understand what properties of neuronal networks promote synchrony of neural firing. One measure that is often used to determine the contribution of individual neurons to network synchrony is called the phase response curve (PRC). PRCs describe how the timing of neuronal firing changes depending on when input, such as a synaptic signal, is received by the neuron. A characteristic of PRCs that has previously not been well understood is that they change dramatically as the neuron's firing frequency is modulated. This effect carries potential significance, since cognitive functions are often associated with specific frequencies of network activity in the brain. We showed computationally that the frequency dependence of PRCs can be explained by the relative timing of ionic membrane currents with respect to the time between spike firings. Our simulations also showed that the frequency dependence of neuronal PRCs leads to frequency-dependent changes in network synchronization that can be different for different neuron types. These results further our understanding of how synchronization is generated in the brain to support various cognitive functions.
doi:10.1371/journal.pcbi.1002062
PMCID: PMC3098201  PMID: 21625571
3.  Impact of Adaptation Currents on Synchronization of Coupled Exponential Integrate-and-Fire Neurons 
PLoS Computational Biology  2012;8(4):e1002478.
The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.
Author Summary
Synchronization of neuronal spiking in the brain is related to cognitive functions, such as perception, attention, and memory. It is therefore important to determine which properties of neurons influence their collective behavior in a network and to understand how. A prominent feature of many cortical neurons is spike frequency adaptation, which is caused by slow transmembrane currents. We investigated how these adaptation currents affect the synchronization tendency of coupled model neurons. Using the efficient adaptive exponential integrate-and-fire (aEIF) model and a biophysically detailed neuron model for validation, we found that increased adaptation currents promote synchronization of coupled excitatory neurons at lower spike frequencies, as long as the conduction delays between the neurons are negligible. Inhibitory neurons on the other hand synchronize in presence of conduction delays, with or without adaptation currents. Our results emphasize the utility of the aEIF model for computational studies of neuronal network dynamics. We conclude that adaptation currents provide a mechanism to generate low frequency oscillations in local populations of excitatory neurons, while faster rhythms seem to be caused by inhibition rather than excitation.
doi:10.1371/journal.pcbi.1002478
PMCID: PMC3325187  PMID: 22511861
4.  Fear conditioning-related changes in cerebellar Purkinje cell activities in goldfish 
Background
Fear conditioning-induced changes in cerebellar Purkinje cell responses to a conditioned stimulus have been reported in rabbits. It has been suggested that synaptic long-term potentiation and the resulting increases in firing rates of Purkinje cells are related to the acquisition of conditioned fear in mammals. However, Purkinje cell activities during acquisition of conditioned fear have not been analysed, and changes in Purkinje cell activities throughout the development of conditioned fear have not yet been investigated. In the present study, we tracked Purkinje cell activities throughout a fear conditioning procedure and aimed to elucidate further how cerebellar circuits function during the acquisition and expression of conditioned fear.
Methods
Activities of single Purkinje cells in the corpus cerebelli were tracked throughout a classical fear conditioning procedure in goldfish. A delayed conditioning paradigm was used with cardiac deceleration as the conditioned response. Conditioning-related changes of Purkinje cell responses to a conditioned stimulus and unconditioned stimulus were examined.
Results
The majority of Purkinje cells sampled responded to the conditioned stimulus by either increasing or decreasing their firing rates before training. Although there were various types of conditioning-related changes in Purkinje cells, more than half of the cells showed suppressed activities in response to the conditioned stimulus after acquisition of conditioned fear. Purkinje cells that showed unconditioned stimulus-coupled complex-spike firings also exhibited conditioning-related suppression of simple-spike responses to the conditioned stimulus. A small number of Purkinje cells showed increased excitatory responses in the acquisition sessions. We found that the magnitudes of changes in the firing frequencies of some Purkinje cells in response to the conditioned stimulus correlated with the magnitudes of the conditioned responses on a trial-to-trial basis.
Conclusions
These results demonstrate that Purkinje cells in the corpus cerebelli of goldfish show fear conditioning-related changes in response to a stimulus that had been emotionally neutral prior to conditioning. Unconditioned stimulus-induced climbing fibre inputs to the Purkinje cells may be involved in mediating these plastic changes.
doi:10.1186/1744-9081-8-52
PMCID: PMC3505750  PMID: 23114007
Cerebellum; Fear conditioning; Goldfish; Purkinje cell
5.  Interaction of plasticity and circuit organization during the acquisition of cerebellum-dependent motor learning 
eLife  2013;2:e01574.
Motor learning occurs through interactions between the cerebellar circuit and cellular plasticity at different sites. Previous work has established plasticity in brain slices and suggested plausible sites of behavioral learning. We now reveal what actually happens in the cerebellum during short-term learning. We monitor the expression of plasticity in the simple-spike firing of cerebellar Purkinje cells during trial-over-trial learning in smooth pursuit eye movements of monkeys. Our findings imply that: 1) a single complex-spike response driven by one instruction for learning causes short-term plasticity in a Purkinje cell’s mossy fiber/parallel-fiber input pathways; 2) complex-spike responses and simple-spike firing rate are correlated across the Purkinje cell population; and 3) simple-spike firing rate at the time of an instruction for learning modulates the probability of a complex-spike response, possibly through a disynaptic feedback pathway to the inferior olive. These mechanisms may participate in long-term motor learning.
DOI: http://dx.doi.org/10.7554/eLife.01574.001
eLife digest
Practice makes perfect in many areas of life, such as playing sport or even just drinking coffee from a cup without spilling any. Our brains can learn and improve these motor skills through trial, error and learning, with such “motor learning” depending on the cerebellum, a part of the brain that helps to coordinate all kinds of movements.
Motor learning is a product of the organization of the cerebellar circuit, which is well understood, and the “plasticity” in the synapses that determine how cerebellar neurons interact with each other. The cerebellum contains cells called Purkinje cells that receive distinctive inputs from two pathways: a pathway involving inputs from many parallel fibers, which convey signals related to sensory events or motor commands; and a pathway involving input from a single climbing-fiber, which conveys signals from a part of the brain called the inferior olive nucleus.
Research on slices of brain has revealed many sites and forms of cerebellar plasticity that could participate in motor learning. In one form of plasticity, the strength of the synapses between the parallel fibers and the Purkinje cell can be changed when a signal sent along the climbing fiber arrives the Purkinje cell.
Yang and Lisberger have now taken the next step by studying the cerebellum of a monkey as it performs a motor learning task. Remarkably these experiments show that the climbing fiber inputs cause plasticity of Purkinje cell activity, just as happens in the experiments on brain slices. Further, some learning in the cerebellum restricts further learning, so that the cerebellum puts boundaries on its own learning. Overall the results make clear how learning is a property of groups of neurons working together in a circuit, rather than simply of changes in the strength of specific synapses.
By shedding light on what happens in the cerebellum during short-term motor learning, the work of Yang and Lisberger will benefit efforts to understand how the cerebellum is involved in motor learning on all time scales.
DOI: http://dx.doi.org/10.7554/eLife.01574.002
doi:10.7554/eLife.01574
PMCID: PMC3871706  PMID: 24381248
non-human primate; smooth pursuit eye movements; climbing fiber; cerebellar learning; trial-over-trial learning; floccular complex; Other
6.  Synchrony and neural coding in cerebellar circuits 
The cerebellum regulates complex movements and is also implicated in cognitive tasks, and cerebellar dysfunction is consequently associated not only with movement disorders, but also with conditions like autism and dyslexia. How information is encoded by specific cerebellar firing patterns remains debated, however. A central question is how the cerebellar cortex transmits its integrated output to the cerebellar nuclei via GABAergic synapses from Purkinje neurons. Possible answers come from accumulating evidence that subsets of Purkinje cells synchronize their firing during behaviors that require the cerebellum. Consistent with models predicting that coherent activity of inhibitory networks has the capacity to dictate firing patterns of target neurons, recent experimental work supports the idea that inhibitory synchrony may regulate the response of cerebellar nuclear cells to Purkinje inputs, owing to the interplay between unusually fast inhibitory synaptic responses and high rates of intrinsic activity. Data from multiple laboratories lead to a working hypothesis that synchronous inhibitory input from Purkinje cells can set the timing and rate of action potentials produced by cerebellar nuclear cells, thereby relaying information out of the cerebellum. If so, then changing spatiotemporal patterns of Purkinje activity would allow different subsets of inhibitory neurons to control cerebellar output at different times. Here we explore the evidence for and against the idea that a synchrony code defines, at least in part, the input–output function between the cerebellar cortex and nuclei. We consider the literature on the existence of simple spike synchrony, convergence of Purkinje neurons onto nuclear neurons, and intrinsic properties of nuclear neurons that contribute to responses to inhibition. Finally, we discuss factors that may disrupt or modulate a synchrony code and describe the potential contributions of inhibitory synchrony to other motor circuits.
doi:10.3389/fncir.2012.00097
PMCID: PMC3518933  PMID: 23248585
Purkinje; cerebellar nuclei; interpositus; corticonuclear; action potential; inhibition; IPSC; spatiotemporal
7.  STD-Dependent and Independent Encoding of Input Irregularity as Spike Rate in a Computational Model of a Cerebellar Nucleus Neuron 
Cerebellum (London, England)  2011;10(4):667-682.
Neurons in the cerebellar nuclei (CN) receive inhibitory inputs from Purkinje cells in the cerebellar cortex and provide the major output from the cerebellum, but their computational function is not well understood. It has recently been shown that the spike activity of Purkinje cells is more regular than previously assumed and that this regularity can affect motor behaviour. We use a conductance-based model of a CN neuron to study the effect of the regularity of Purkinje cell spiking on CN neuron activity. We find that increasing the irregularity of Purkinje cell activity accelerates the CN neuron spike rate and that the mechanism of this recoding of input irregularity as output spike rate depends on the number of Purkinje cells converging onto a CN neuron. For high convergence ratios, the irregularity induced spike rate acceleration depends on short-term depression (STD) at the Purkinje cell synapses. At low convergence ratios, or for synchronised Purkinje cell input, the firing rate increase is independent of STD. The transformation of input irregularity into output spike rate occurs in response to artificial input spike trains as well as to spike trains recorded from Purkinje cells in tottering mice, which show highly irregular spiking patterns. Our results suggest that STD may contribute to the accelerated CN spike rate in tottering mice and they raise the possibility that the deficits in motor control in these mutants partly result as a pathological consequence of this natural form of plasticity.
Electronic supplementary material
The online version of this article (doi:10.1007/s12311-011-0295-9) contains supplementary material, which is available to authorized users.
doi:10.1007/s12311-011-0295-9
PMCID: PMC3215884  PMID: 21761198
Cerebellar nuclei; Purkinje cell; Short-term depression; Tottering; Ataxia
8.  Purkinje neuron synchrony elicits time-locked spiking in the cerebellar nuclei 
Nature  2011;481(7382):502-505.
An unusual feature of the cerebellar cortex is that its output neurons, Purkinje cells, are GABAergic. Their high intrinsic firing rates1 (50 Hz) and extensive convergence2,3 predict that that target neurons in the cerebellar nuclei would be largely inhibited unless Purkinje cells pause their spiking, yet Purkinje and nuclear neuron firing rates do not always vary inversely4. A potential clue to how these synapses transmit information is that populations of Purkinje neurons synchronize their spikes during cerebellar behaviors5–11. If nuclear neurons respond to Purkinje synchrony, they may encode signals from subsets of inhibitory inputs7,12–14. Here we show in weanling and adult mice that nuclear neurons transmit the timing of synchronous Purkinje afferent spikes, owing to modest Purkinje-to-nuclear convergence ratios (~40:1), fast IPSC kinetics (τdecay=2.5 ms), and high intrinsic firing rates (~90 Hz). In vitro, dynamically clamped asynchronous IPSPs mimicking Purkinje afferents suppress nuclear cell spiking, whereas synchronous IPSPs entrain nuclear cell spiking. With partial synchrony, nuclear neurons time-lock their spikes to the synchronous subpopulation of inputs, even when only 2 of 40 afferents synchronize. In vivo, nuclear neurons reliably phase-lock to regular trains of molecular layer stimulation. Thus, cerebellar nuclear neurons can preferentially relay the spike timing of synchronized Purkinje cells to downstream premotor areas.
doi:10.1038/nature10732
PMCID: PMC3268051  PMID: 22198670
9.  Changes in complex spike activity during classical conditioning 
The cerebellar cortex is necessary for adaptively timed conditioned responses (CRs) in eyeblink conditioning. During conditioning, Purkinje cells acquire pause responses or “Purkinje cell CRs” to the conditioned stimuli (CS), resulting in disinhibition of the cerebellar nuclei (CN), allowing them to activate motor nuclei that control eyeblinks. This disinhibition also causes inhibition of the inferior olive (IO), via the nucleo-olivary pathway (N-O). Activation of the IO, which relays the unconditional stimulus (US) to the cortex, elicits characteristic complex spikes in Purkinje cells. Although Purkinje cell activity, as well as stimulation of the CN, is known to influence IO activity, much remains to be learned about the way that learned changes in simple spike firing affects the IO. In the present study, we analyzed changes in simple and complex spike firing, in extracellular Purkinje cell records, from the C3 zone, in decerebrate ferrets undergoing training in a conditioning paradigm. In agreement with the N-O feedback hypothesis, acquisition resulted in a gradual decrease in complex spike activity during the conditioned stimulus, with a delay that is consistent with the long N-O latency. Also supporting the feedback hypothesis, training with a short interstimulus interval (ISI), which does not lead to acquisition of a Purkinje cell CR, did not cause a suppression of complex spike activity. In contrast, observations that extinction did not lead to a recovery in complex spike activity and the irregular patterns of simple and complex spike activity after the conditioned stimulus are less conclusive.
doi:10.3389/fncir.2014.00090
PMCID: PMC4122208  PMID: 25140129
inferior olive; nucleo-olivary pathway; complex spikes; purkinje cells; in vivo electrophysiology; eyeblink conditioning; oscillations; interstimulus interval
10.  Patterns of interval correlations in neural oscillators with adaptation 
Neural firing is often subject to negative feedback by adaptation currents. These currents can induce strong correlations among the time intervals between spikes. Here we study analytically the interval correlations of a broad class of noisy neural oscillators with spike-triggered adaptation of arbitrary strength and time scale. Our weak-noise theory provides a general relation between the correlations and the phase-response curve (PRC) of the oscillator, proves anti-correlations between neighboring intervals for adapting neurons with type I PRC and identifies a single order parameter that determines the qualitative pattern of correlations. Monotonically decaying or oscillating correlation structures can be related to qualitatively different voltage traces after spiking, which can be explained by the phase plane geometry. At high firing rates, the long-term variability of the spike train associated with the cumulative interval correlations becomes small, independent of model details. Our results are verified by comparison with stochastic simulations of the exponential, leaky, and generalized integrate-and-fire models with adaptation.
doi:10.3389/fncom.2013.00164
PMCID: PMC3843362  PMID: 24348372
spike-frequency adaptation; non-renewal process; serial correlation coefficient; phase-response curve; integrate-and-fire model; long-term variability
11.  Rapid development of Purkinje cell excitability, functional cerebellar circuit, and afferent sensory input to cerebellum in zebrafish 
The zebrafish has significant advantages for studying the morphological development of the brain. However, little is known about the functional development of the zebrafish brain. We used patch clamp electrophysiology in live animals to investigate the emergence of excitability in cerebellar Purkinje cells, functional maturation of the cerebellar circuit, and establishment of sensory input to the cerebellum. Purkinje cells are born at 3 days post-fertilization (dpf). By 4 dpf, Purkinje cells spontaneously fired action potentials in an irregular pattern. By 5 dpf, the frequency and regularity of tonic firing had increased significantly and most cells fired complex spikes in response to climbing fiber activation. Our data suggest that, as in mammals, Purkinje cells are initially innervated by multiple climbing fibers that are winnowed to a single input. To probe the development of functional sensory input to the cerebellum, we investigated the response of Purkinje cells to a visual stimulus consisting of a rapid change in light intensity. At 4 dpf, sudden darkness increased the rate of tonic firing, suggesting that afferent pathways carrying visual information are already active by this stage. By 5 dpf, visual stimuli also activated climbing fibers, increasing the frequency of complex spiking. Our results indicate that the electrical properties of zebrafish and mammalian Purkinje cells are highly conserved and suggest that the same ion channels, Nav1.6 and Kv3.3, underlie spontaneous pacemaking activity. Interestingly, functional development of the cerebellum is temporally correlated with the emergence of complex, visually-guided behaviors such as prey capture. Because of the rapid formation of an electrically-active cerebellum, optical transparency, and ease of genetic manipulation, the zebrafish has great potential for functionally mapping cerebellar afferent and efferent pathways and for investigating cerebellar control of motor behavior.
doi:10.3389/fncir.2014.00147
PMCID: PMC4271617  PMID: 25565973
Purkinje cell; patch clamp; cerebellum; zebrafish; parallel fiber; climbing fiber; visual input
12.  Modeling the Generation of Output by the Cerebellar Nuclei 
Functional aspects of network integration in the cerebellar cortex have been studied experimentally and modeled in much detail ever since the early work by theoreticians such as Marr, Albus and Braitenberg more than 40 years ago. In contrast, much less is known about cerebellar processing at the output stage, namely in the cerebellar nuclei (CN). Here, input from Purkinje cells converges to control CN neuron spiking via GABAergic inhibition, before the output from the CN reaches cerebellar targets such as the brainstem and the motor thalamus. In this article we review modeling studies that address how the CN may integrate cerebellar cortical inputs, and what kind of signals may be transmitted. Specific hypotheses in the literature contrast rate coding and temporal coding of information in the spiking output from the CN. One popular hypothesis states that postinhibitory rebound spiking may be an important mechanism by which Purkinje cell inhibition is turned into CN output spiking, but this hypothesis remains controversial. Rate coding clearly does take place, but in what way it may be augmented by temporal codes remains to be more clearly established. Several candidate mechanisms distinct from rebound spiking are discussed, such as the significance of spike time correlations between Purkinje cell pools to determine CN spike timing, irregularity of Purkinje cell spiking as a determinant of CN firing rate, and shared brief pauses between Purkinje cell pools that may trigger individual CN spikes precisely.
doi:10.1016/j.neunet.2012.11.006
PMCID: PMC3596440  PMID: 23200193
Purkinje cell; neural coding; STD; rebound; compartmental model; dynamic clamp
13.  Effect of phase response curve skew on synchronization with and without conduction delays 
A central problem in cortical processing including sensory binding and attentional gating is how neurons can synchronize their responses with zero or near-zero time lag. For a spontaneously firing neuron, an input from another neuron can delay or advance the next spike by different amounts depending upon the timing of the input relative to the previous spike. This information constitutes the phase response curve (PRC). We present a simple graphical method for determining the effect of PRC shape on synchronization tendencies and illustrate it using type 1 PRCs, which consist entirely of advances (delays) in response to excitation (inhibition). We obtained the following generic solutions for type 1 PRCs, which include the pulse-coupled leaky integrate and fire model. For pairs with mutual excitation, exact synchrony can be stable for strong coupling because of the stabilizing effect of the causal limit region of the PRC in which an input triggers a spike immediately upon arrival. However, synchrony is unstable for short delays, because delayed inputs arrive during a refractory period and cannot trigger an immediate spike. Right skew destabilizes antiphase and enables modes with time lags that grow as the conduction delay is increased. Therefore, right skew favors near synchrony at short conduction delays and a gradual transition between synchrony and antiphase for pairs coupled by mutual excitation. For pairs with mutual inhibition, zero time lag synchrony is stable for conduction delays ranging from zero to a substantial fraction of the period for pairs. However, for right skew there is a preferred antiphase mode at short delays. In contrast to mutual excitation, left skew destabilizes antiphase for mutual inhibition so that synchrony dominates at short delays as well. These pairwise synchronization tendencies constrain the synchronization properties of neurons embedded in larger networks.
doi:10.3389/fncir.2013.00194
PMCID: PMC3858834  PMID: 24376399
synchrony; synchronization; pulsatile coupling; phase locking; phase resetting
14.  Force sensor in simulated skin and neural model mimic tactile SAI afferent spiking response to ramp and hold stimuli 
Background
The next generation of prosthetic limbs will restore sensory feedback to the nervous system by mimicking how skin mechanoreceptors, innervated by afferents, produce trains of action potentials in response to compressive stimuli. Prior work has addressed building sensors within skin substitutes for robotics, modeling skin mechanics and neural dynamics of mechanotransduction, and predicting response timing of action potentials for vibration. The effort here is unique because it accounts for skin elasticity by measuring force within simulated skin, utilizes few free model parameters for parsimony, and separates parameter fitting and model validation. Additionally, the ramp-and-hold, sustained stimuli used in this work capture the essential features of the everyday task of contacting and holding an object.
Methods
This systems integration effort computationally replicates the neural firing behavior for a slowly adapting type I (SAI) afferent in its temporally varying response to both intensity and rate of indentation force by combining a physical force sensor, housed in a skin-like substrate, with a mathematical model of neuronal spiking, the leaky integrate-and-fire. Comparison experiments were then conducted using ramp-and-hold stimuli on both the spiking-sensor model and mouse SAI afferents. The model parameters were iteratively fit against recorded SAI interspike intervals (ISI) before validating the model to assess its performance.
Results
Model-predicted spike firing compares favorably with that observed for single SAI afferents. As indentation magnitude increases (1.2, 1.3, to 1.4 mm), mean ISI decreases from 98.81 ± 24.73, 54.52 ± 6.94, to 41.11 ± 6.11 ms. Moreover, as rate of ramp-up increases, ISI during ramp-up decreases from 21.85 ± 5.33, 19.98 ± 3.10, to 15.42 ± 2.41 ms. Considering first spikes, the predicted latencies exhibited a decreasing trend as stimulus rate increased, as is observed in afferent recordings. Finally, the SAI afferent’s characteristic response of producing irregular ISIs is shown to be controllable via manipulating the output filtering from the sensor or adding stochastic noise.
Conclusions
This integrated engineering approach extends prior works focused upon neural dynamics and vibration. Future efforts will perfect measures of performance, such as first spike latency and irregular ISIs, and link the generation of characteristic features within trains of action potentials with current pulse waveforms that stimulate single action potentials at the peripheral afferent.
doi:10.1186/1743-0003-9-45
PMCID: PMC3506479  PMID: 22824523
Tactile; Force sensor; Leaky integrate-and-fire; Mechanoreceptor; SAI; Neural; Prosthetic; Biomechanics; Electrophysiology; Skin; Elasticity
15.  Phase-response curves and synchronized neural networks 
We review the principal assumptions underlying the application of phase-response curves (PRCs) to synchronization in neuronal networks. The PRC measures how much a given synaptic input perturbs spike timing in a neural oscillator. Among other applications, PRCs make explicit predictions about whether a given network of interconnected neurons will synchronize, as is often observed in cortical structures. Regarding the assumptions of the PRC theory, we conclude: (i) The assumption of noise-tolerant cellular oscillations at or near the network frequency holds in some but not all cases. (ii) Reduced models for PRC-based analysis can be formally related to more realistic models. (iii) Spike-rate adaptation limits PRC-based analysis but does not invalidate it. (iv) The dependence of PRCs on synaptic location emphasizes the importance of improving methods of synaptic stimulation. (v) New methods can distinguish between oscillations that derive from mutual connections and those arising from common drive. (vi) It is helpful to assume linear summation of effects of synaptic inputs; experiments with trains of inputs call this assumption into question. (vii) Relatively subtle changes in network structure can invalidate PRC-based predictions. (viii) Heterogeneity in the preferred frequencies of component neurons does not invalidate PRC analysis, but can annihilate synchronous activity.
doi:10.1098/rstb.2009.0292
PMCID: PMC2894948  PMID: 20603361
neural network; phase-response curve; computational neuroscience
16.  Regular Patterns in Cerebellar Purkinje Cell Simple Spike Trains 
PLoS ONE  2007;2(5):e485.
Background
Cerebellar Purkinje cells (PC) in vivo are commonly reported to generate irregular spike trains, documented by high coefficients of variation of interspike-intervals (ISI). In strong contrast, they fire very regularly in the in vitro slice preparation. We studied the nature of this difference in firing properties by focusing on short-term variability and its dependence on behavioral state.
Methodology/Principal Findings
Using an analysis based on CV2 values, we could isolate precise regular spiking patterns, lasting up to hundreds of milliseconds, in PC simple spike trains recorded in both anesthetized and awake rodents. Regular spike patterns, defined by low variability of successive ISIs, comprised over half of the spikes, showed a wide range of mean ISIs, and were affected by behavioral state and tactile stimulation. Interestingly, regular patterns often coincided in nearby Purkinje cells without precise synchronization of individual spikes. Regular patterns exclusively appeared during the up state of the PC membrane potential, while single ISIs occurred both during up and down states. Possible functional consequences of regular spike patterns were investigated by modeling the synaptic conductance in neurons of the deep cerebellar nuclei (DCN). Simulations showed that these regular patterns caused epochs of relatively constant synaptic conductance in DCN neurons.
Conclusions/Significance
Our findings indicate that the apparent irregularity in cerebellar PC simple spike trains in vivo is most likely caused by mixing of different regular spike patterns, separated by single long intervals, over time. We propose that PCs may signal information, at least in part, in regular spike patterns to downstream DCN neurons.
doi:10.1371/journal.pone.0000485
PMCID: PMC1868782  PMID: 17534435
17.  Robustness, variability, phase dependence, and longevity of individual synaptic input effects on spike timing during fluctuating synaptic backgrounds: A modeling study of globus pallidus neuron phase response properties 
Neuroscience  2012;219:92-110.
A neuron’s phase response curve (PRC) shows how inputs arriving at different times during the spike cycle differentially affect the timing of subsequent spikes. Using a full morphological model of a globus pallidus (GP) neuron, we previously demonstrated that dendritic conductances shape the PRC in a spike frequency dependent manner, suggesting different functional roles of perisomatic and distal dendritic synapses in the control of patterned network activity. In the present study we extend this analysis to examine the impact of physiologically realistic high conductance states on somatic and dendritic PRCs and the time course of spike train perturbations. First, we found that average somatic and dendritic PRCs preserved their shapes and spike frequency dependence when the model was driven by spatially-distributed, stochastic conductance inputs rather than tonic somatic current. However, responses to inputs during specific synaptic backgrounds often deviated substantially from the average PRC. Therefore, we analyzed the interactions of PRC stimuli with transient fluctuations in the synaptic background on a trial-by-trial basis. We found that the variability in responses to PRC stimuli and the incidence of stimulus-evoked added or skipped spikes were stimulus-phase-dependent and reflected the profile of the average PRC, suggesting commonality in the underlying mechanisms. Clear differences in the relation between the phase of input and variability of spike response between dendritic and somatic inputs indicate that theses regions generally represent distinct dynamical subsystems of synaptic integration with respect to influencing the stability of spike time attractors generated by the overall synaptic conductance.
doi:10.1016/j.neuroscience.2012.05.059
PMCID: PMC3402697  PMID: 22659567
phase response curve (PRC); high conductance state; stochastic synaptic background; spike time attractor; dendrite; SK current; synchronization; oscillation
18.  Pulse Coupled Oscillators and the Phase Resetting Curve 
Mathematical biosciences  2010;226(2):77-96.
Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.
doi:10.1016/j.mbs.2010.05.001
PMCID: PMC3022482  PMID: 20460132
Pulse coupled oscillators; Phase resetting; Phase locking; Synchronization; Splay; Clustering
19.  How Noisy Adaptation of Neurons Shapes Interspike Interval Histograms and Correlations 
PLoS Computational Biology  2010;6(12):e1001026.
Channel noise is the dominant intrinsic noise source of neurons causing variability in the timing of action potentials and interspike intervals (ISI). Slow adaptation currents are observed in many cells and strongly shape response properties of neurons. These currents are mediated by finite populations of ionic channels and may thus carry a substantial noise component. Here we study the effect of such adaptation noise on the ISI statistics of an integrate-and-fire model neuron by means of analytical techniques and extensive numerical simulations. We contrast this stochastic adaptation with the commonly studied case of a fast fluctuating current noise and a deterministic adaptation current (corresponding to an infinite population of adaptation channels). We derive analytical approximations for the ISI density and ISI serial correlation coefficient for both cases. For fast fluctuations and deterministic adaptation, the ISI density is well approximated by an inverse Gaussian (IG) and the ISI correlations are negative. In marked contrast, for stochastic adaptation, the density is more peaked and has a heavier tail than an IG density and the serial correlations are positive. A numerical study of the mixed case where both fast fluctuations and adaptation channel noise are present reveals a smooth transition between the analytically tractable limiting cases. Our conclusions are furthermore supported by numerical simulations of a biophysically more realistic Hodgkin-Huxley type model. Our results could be used to infer the dominant source of noise in neurons from their ISI statistics.
Author Summary
Neurons of sensory systems encode signals from the environment by sequences of electrical pulses — so-called spikes. This coding of information is fundamentally limited by the presence of intrinsic neural noise. One major noise source is “channel noise” that is generated by the random activity of various types of ion channels in the cell membrane. Slow adaptation currents can also be a source of channel noise. Adaptation currents profoundly shape the signal transmission properties of a neuron by emphasizing fast changes in the stimulus but adapting the spiking frequency to slow stimulus components. Here, we mathematically analyze the effects of both slow adaptation channel noise and fast channel noise on the statistics of spike times in adapting neuron models. Surprisingly, the two noise sources result in qualitatively different distributions and correlations of time intervals between spikes. Our findings add a novel aspect to the function of adaptation currents and can also be used to experimentally distinguish adaptation noise and fast channel noise on the basis of spike sequences.
doi:10.1371/journal.pcbi.1001026
PMCID: PMC3002986  PMID: 21187900
20.  Coding of stimulus strength via analog calcium signals in Purkinje cell dendrites of awake mice 
eLife  2014;3:e03663.
The climbing fiber input to Purkinje cells acts as a teaching signal by triggering a massive influx of dendritic calcium that marks the occurrence of instructive stimuli during cerebellar learning. Here, we challenge the view that these calcium spikes are all-or-none and only signal whether the instructive stimulus has occurred, without providing parametric information about its features. We imaged ensembles of Purkinje cell dendrites in awake mice and measured their calcium responses to periocular airpuffs that serve as instructive stimuli during cerebellar-dependent eyeblink conditioning. Information about airpuff duration and pressure was encoded probabilistically across repeated trials, and in two additional signals in single trials: the synchrony of calcium spikes in the Purkinje cell population, and the amplitude of the calcium spikes, which was modulated by a non-climbing fiber pathway. These results indicate that calcium-based teaching signals in Purkinje cells contain analog information that encodes the strength of instructive stimuli trial-by-trial.
DOI: http://dx.doi.org/10.7554/eLife.03663.001
eLife digest
A region of the brain known as the cerebellum plays a key role in learning how to anticipate an event. For example, if you know that a puff of air is going to be directed at your eye, it's a good idea to close it in advance. However, how much you need to close it depends on how strong that puff of air is. A very strong puff might require closing the eye completely to protect it. In contrast, it is probably better to only partially close the eye if you know a lighter puff of air is coming, so that you can still see.
Extensive research has focused on how neurons in and around the cerebellum work together to achieve this goal. When an event—such as a puff of air—occurs, signals are sent to large neurons in the cerebellum, called Purkinje cells, by ‘climbing fibers’. However, climbing fibers were thought to be able to respond in only two ways: either they fire in a single burst to signal that an event has occurred, or they don't fire. It was therefore unclear how the finer details of the event (for example, the strength of the puff of air) are transmitted to the cerebellum.
Najafi et al. imaged the level of calcium in the cerebellum of mice, as this indicates how active the neurons are. When a puff of air was directed at the eyes of the mice, Najafi et al. saw that the size of the response of the Purkinje cells corresponded with how big the puff of air was. Najafi et al. show that the size of this response, which is based mostly on input from the climbing fibers, is also influenced by input from an additional unknown source.
These findings show that Purkinje cells of the cerebellum receive detailed information about the nature of an event, such as a puff of air. What remains to be seen is whether the cerebellum uses this information to learn the correct response, that is how hard to blink to avoid the expected puff.
DOI: http://dx.doi.org/10.7554/eLife.03663.002
doi:10.7554/eLife.03663
PMCID: PMC4158287  PMID: 25205669
cerebellum; plasticity; climbing fiber; motor learning; unconditioned stimulus; neural coding; mouse
21.  Short Conduction Delays Cause Inhibition Rather than Excitation to Favor Synchrony in Hybrid Neuronal Networks of the Entorhinal Cortex 
PLoS Computational Biology  2012;8(1):e1002306.
How stable synchrony in neuronal networks is sustained in the presence of conduction delays is an open question. The Dynamic Clamp was used to measure phase resetting curves (PRCs) for entorhinal cortical cells, and then to construct networks of two such neurons. PRCs were in general Type I (all advances or all delays) or weakly type II with a small region at early phases with the opposite type of resetting. We used previously developed theoretical methods based on PRCs under the assumption of pulsatile coupling to predict the delays that synchronize these hybrid circuits. For excitatory coupling, synchrony was predicted and observed only with no delay and for delays greater than half a network period that cause each neuron to receive an input late in its firing cycle and almost immediately fire an action potential. Synchronization for these long delays was surprisingly tight and robust to the noise and heterogeneity inherent in a biological system. In contrast to excitatory coupling, inhibitory coupling led to antiphase for no delay, very short delays and delays close to a network period, but to near-synchrony for a wide range of relatively short delays. PRC-based methods show that conduction delays can stabilize synchrony in several ways, including neutralizing a discontinuity introduced by strong inhibition, favoring synchrony in the case of noisy bistability, and avoiding an initial destabilizing region of a weakly type II PRC. PRCs can identify optimal conduction delays favoring synchronization at a given frequency, and also predict robustness to noise and heterogeneity.
Author Summary
Individual oscillators, such as pendulum-based clocks and fireflies, can spontaneously organize into a coherent, synchronized entity with a common frequency. Neurons can oscillate under some circumstances, and can synchronize their firing both within and across brain regions. Synchronized assemblies of neurons are thought to underlie cognitive functions such as recognition, recall, perception and attention. Pathological synchrony can lead to epilepsy, tremor and other dynamical diseases, and synchronization is altered in most mental disorders. Biological neurons synchronize despite conduction delays, heterogeneous circuit composition, and noise. In biological experiments, we built simple networks in which two living neurons could interact via a computer in real time. The computer precisely controlled the nature of the connectivity and the length of the communication delays. We characterized the synchronization tendencies of individual, isolated oscillators by measuring how much a single input delivered by the computer transiently shortened or lengthened the cycle period of the oscillation. We then used this information to correctly predict the strong dependence of the coordination pattern of the firing of the component neurons on the length of the communication delays. Upon this foundation, we can begin to build a theory of the basic principles of synchronization in more complex brain circuits.
doi:10.1371/journal.pcbi.1002306
PMCID: PMC3252263  PMID: 22241969
22.  Cerebellar Zonal Patterning Relies on Purkinje Cell Neurotransmission 
The Journal of Neuroscience  2014;34(24):8231-8245.
Cerebellar circuits are patterned into an array of topographic parasagittal domains called zones. The proper connectivity of zones is critical for motor coordination and motor learning, and in several neurological diseases cerebellar circuits degenerate in zonal patterns. Despite recent advances in understanding zone function, we still have a limited understanding of how zones are formed. Here, we focused our attention on Purkinje cells to gain a better understanding of their specific role in establishing zonal circuits. We used conditional mouse genetics to test the hypothesis that Purkinje cell neurotransmission is essential for refining prefunctional developmental zones into sharp functional zones. Our results show that inhibitory synaptic transmission in Purkinje cells is necessary for the precise patterning of Purkinje cell zones and the topographic targeting of mossy fiber afferents. As expected, blocking Purkinje cell neurotransmission caused ataxia. Using in vivo electrophysiology, we demonstrate that loss of Purkinje cell communication altered the firing rate and pattern of their target cerebellar nuclear neurons. Analysis of Purkinje cell complex spike firing revealed that feedback in the cerebellar nuclei to inferior olive to Purkinje cell loop is obstructed. Loss of Purkinje neurotransmission also caused ectopic zonal expression of tyrosine hydroxylase, which is only expressed in adult Purkinje cells when calcium is dysregulated and if excitability is altered. Our results suggest that Purkinje cell inhibitory neurotransmission establishes the functional circuitry of the cerebellum by patterning the molecular zones, fine-tuning afferent circuitry, and shaping neuronal activity.
doi:10.1523/JNEUROSCI.0122-14.2014
PMCID: PMC4051975  PMID: 24920627
ataxia; circuitry; connectivity; gene expression; inhibition; physiology
23.  The Output Signal of Purkinje Cells of the Cerebellum and Circadian Rhythmicity 
PLoS ONE  2013;8(3):e58457.
Measurement of clock gene expression has recently provided evidence that the cerebellum, like the master clock in the SCN, contains a circadian oscillator. The cerebellar oscillator is involved in anticipation of mealtime and possibly resides in Purkinje cells. However, the rhythmic gene expression is likely transduced into a circadian cerebellar output signal to exert an effective control of neuronal brain circuits that are responsible for feeding behavior. Using electrophysiological recordings from acute and organotypic cerebellar slices, we tested the hypothesis whether Purkinje cells transmit a circadian modulated signal to their targets in the brain. Extracellular recordings from brain slices revealed the typical discharge pattern previously described in vivo in single cell recordings showing basically a tonic or a trimodal-like firing pattern. However, in acute sagittal cerebellar slices the average spike rate of randomly selected Purkinje cells did not exhibit significant circadian variations, irrespective of their specific firing pattern. Also, frequency and amplitude of spontaneous inhibitory postsynaptic currents and the amplitude of GABA- and glutamate-evoked currents did not vary with circadian time. Long-term recordings using multielectrode arrays (MEA) allowed to monitor neuronal activity at multiple sites in organotypic cerebellar slices for several days to weeks. With this recording technique we observed oscillations of the firing rate of cerebellar neurons, presumably of Purkinje cells, with a period of about 24 hours which were stable for periods up to three days. The daily renewal of culture medium could induce circadian oscillations of the firing rate of Purkinje cells, a feature that is compatible with the behavior of slave oscillators. However, from the present results it appears that the circadian expression of cerebellar clock genes exerts only a weak influence on the electrical output of cerebellar neurons.
doi:10.1371/journal.pone.0058457
PMCID: PMC3591352  PMID: 23505510
24.  Role of Plasticity at Different Sites across the Time Course of Cerebellar Motor Learning 
The Journal of Neuroscience  2014;34(21):7077-7090.
Learning comprises multiple components that probably involve cellular and synaptic plasticity at multiple sites. Different neural sites may play their largest roles at different times during behavioral learning. We have used motor learning in smooth pursuit eye movements of monkeys to determine how and when different components of learning occur in a known cerebellar circuit. The earliest learning occurs when one climbing-fiber response to a learning instruction causes simple-spike firing rate of Purkinje cells in the floccular complex of the cerebellum to be depressed transiently at the time of the instruction on the next trial. Trial-over-trial depression and the associated learning in eye movement are forgotten in <6 s, but facilitate long-term behavioral learning over a time scale of ∼5 min. During 100 repetitions of a learning instruction, simple-spike firing rate becomes progressively depressed in Purkinje cells that receive climbing-fiber inputs from the instruction. In Purkinje cells that prefer the opposite direction of pursuit and therefore do not receive climbing-fiber inputs related to the instruction, simple-spike responses undergo potentiation, but more weakly and more slowly. Analysis of the relationship between the learned changes in simple-spike firing and learning in eye velocity suggests an orderly progression of plasticity: first on Purkinje cells with complex-spike (CS) responses to the instruction, later on Purkinje cells with CS responses to the opposite direction of instruction, and last in sites outside the cerebellar cortex. Climbing-fiber inputs appear to play a fast and primary, but nonexclusive, role in pursuit learning.
doi:10.1523/JNEUROSCI.0017-14.2014
PMCID: PMC4028490  PMID: 24849344
climbing fibers; floccular complex; long-term depression; smooth pursuit eye movement; synaptic plasticity; trial-over-trial learning
25.  Stimulus Discrimination in Cerebellar Purkinje Neurons 
PLoS ONE  2014;9(2):e87828.
Cerebellar Purkinje neurons fire spontaneously in the absence of synaptic input. Overlaid on this intrinsic activity, excitatory input from parallel fibres can add simple spikes to the output train, whereas inhibitory input from interneurons can introduce pauses. These and other influences lead to an irregular spike train output in Purkinje neurons in vitro and in vivo, supplying a variable inhibitory drive to deep cerebellar nuclear neurons. From a computational perspective, this variability raises some questions, as individual spikes induced by excitatory inputs are indistinguishable from intrinsic firing activity. Although bursts of high-frequency excitatory input could be discriminated unambiguously from background activity, granule neurons are known to fire in vivo over a wide range of frequencies. This would mean that much of the sensory information relayed through the cerebellar cortex would be lost within the random variation in background activity. We speculated that alternative mechanisms for signal discrimination may exist, and sought to identify characteristic motifs within the sequence of spikes that followed stimulation events. We found that under certain conditions, parallel fibre stimulation could reliably add a “couplet” of spikes with an unusually short interspike interval to the output train. Therefore, despite representing a small fraction of the total number of spikes, these signals can be reliably discriminated from background firing on a moment-to-moment basis, and could result in a differential downstream response.
doi:10.1371/journal.pone.0087828
PMCID: PMC3914862  PMID: 24505320

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