Predicting the responses of repetitively firing neurons to current noise

Entrainment by transcranial alternating current stimulation: Insights from models of cortical oscillations and dynamical systems theory

Signature of neuronal oscillations can be found in nearly every brain function. However, abnormal oscillatory activity is linked with several brain disorders. Transcranial alternating current stimulation (tACS) is a non-invasive brain stimulation technique that can potentially modulate neuronal oscillations and influence behavior both in health and disease. Yet, a complete understanding of how interacting networks of neurons are affected by tACS remains elusive. Entrainment effects by which tACS synchronizes neuronal oscillations is one of the main hypothesized mechanisms, as evidenced in animals and humans. Computational models of cortical oscillations may shed light on the entrainment effects of tACS, but current modeling studies lack specific guidelines to inform experimental investigations. This study addresses the existing gap in understanding the mechanisms of tACS effects on rhythmogenesis within the brain by providing a comprehensive overview of both theoretical and experimental perspectives. We explore the intricate interactions between oscillators and periodic stimulation through the lens of dynamical systems theory. Subsequently, we present a synthesis of experimental findings that demonstrate the effects of tACS on both individual neurons and collective oscillatory patterns in animal models and humans. Our review extends to computational investigations that elucidate the interplay between tACS and neuronal dynamics across diverse cortical network models. To illustrate these concepts, we conclude with a simple oscillatory neuron model, showcasing how fundamental theories of oscillatory behavior derived from dynamical systems, such as phase response of neurons to external perturbation, can account for the entrainment effects observed with tACS. Studies reviewed here render the necessity of integrated experimental and computational approaches for effective neuromodulation by tACS in health and disease.

Impact of Unitary Synaptic Inhibition on Spike Timing in Ventral Tegmental Area Dopamine Neurons

Midbrain dopamine neurons receive convergent synaptic input from multiple brain areas, which perturbs rhythmic pacemaking to produce the complex firing patterns observed in vivo. This study investigated the impact of single and multiple inhibitory inputs on ventral tegmental area (VTA) dopamine neuron firing in mice of both sexes using novel experimental measurements and modeling. We first measured unitary inhibitory postsynaptic currents produced by single axons using both minimal electrical stimulation and minimal optical stimulation of rostromedial tegmental nucleus and ventral pallidum afferents. We next determined the phase resetting curve, the reversal potential for GABAA receptor-mediated inhibitory postsynaptic currents (IPSCs), and the average interspike membrane potential trajectory during pacemaking. We combined these data in a phase oscillator model of a VTA dopamine neuron, simulating the effects of unitary inhibitory postsynaptic conductances (uIPSGs) on spike timing and rate. The effect of a uIPSG on spike timing was predicted to vary according to its timing within the interspike interval or phase. Simulations were performed to predict the pause duration resulting from the synchronous arrival of multiple uIPSGs and the changes in firing rate and regularity produced by asynchronous uIPSGs. The model data suggest that asynchronous inhibition is more effective than synchronous inhibition, because it tends to hold the neuron at membrane potentials well positive to the IPSC reversal potential. Our results indicate that small fluctuations in the inhibitory synaptic input arriving from the many afferents to each dopamine neuron are sufficient to produce highly variable firing patterns, including pauses that have been implicated in reinforcement.

Optogenetic Interrogation of Electrophysiological Dendritic Properties and Their Effect on Pacemaking Neurons from Acute Rodent Brain Slices

Understanding dendritic excitability is essential for a complete and precise characterization of neurons' input-output relationships. Theoretical and experimental work demonstrates that the electrotonic and nonlinear properties of dendrites can alter the amplitude (e.g., through amplification) and latency of synaptic inputs as viewed in the axosomatic region where spike timing is determined. The gold-standard technique to study dendritic excitability is using dual-patch recordings with a high-resistance electrode used to patch a piece of distal dendrite in addition to a somatic patch electrode. However, this approach is often impractical when distal dendrites are too fine to patch. Therefore, we developed a technique that utilizes the expression of Channelrhodopsin-2 (ChR2) to study dendritic excitability in acute brain slices through the combination of a somatic patch electrode and optogenetic activation. The protocol describes how to prepare acute slices from mice that express ChR2 in specific cell types, and how to use two modes of light stimulation: proximal (which activates the soma and proximal dendrites in a ~100 µm diameter surrounding the soma) with the use of a high-magnification objective and full-field stimulation through a low-magnification objective (which activates the entire somato-dendritic field of the neuron). We use this technique in conjunction with various stimulation protocols to estimate model-based spectral components of dendritic filtering and the impact of dendrites on phase response curves, peri-stimulus time histograms, and entrainment of pacemaking neurons. This technique provides a novel use of optogenetics to study intrinsic dendritic excitability through the use of standard patch-clamp slice physiology. Key features • A method for studying the effects of electrotonic and nonlinear dendritic properties on the sub- and suprathreshold responses of pacemaking neurons. • Combines somatic patch clamp or perforated patch recordings with optogenetic activation in acute brain slices to investigate dendritic linear transformation without patching the dendrite. • Oscillatory illumination at various frequencies estimates spectral properties of the dendrite using subthreshold voltage-clamp recordings and studies entrainment of pacemakers in current clamp recordings. • This protocol uses Poisson white noise illumination to estimate dendritic phase response curves and peri-stimulus time histograms.

Phase Delays between Mouse Globus Pallidus Neurons Entrained by Common Oscillatory Drive Arise from Their Intrinsic Properties, Not Their Coupling

A hallmark of Parkinson's disease is the appearance of correlated oscillatory discharge throughout the cortico-basal ganglia (BG) circuits. In the primate globus pallidus (GP), where the discharge of GP neurons is normally uncorrelated, pairs of GP neurons exhibit oscillatory spike correlations with a broad distribution of pairwise phase delays in experimental parkinsonism. The transition to oscillatory correlations is thought to indicate the collapse of the normally segregated information channels traversing the BG. The large phase delays are thought to reflect pathological changes in synaptic connectivity in the BG. Here we study the structure and phase delays of spike correlations measured from neurons in the mouse external GP (GPe) subjected to identical 1-100 Hz sinusoidal drive but recorded in separate experiments. First, we found that spectral modes of a GPe neuron's empirical instantaneous phase response curve (iPRC) elucidate at what phases of the oscillatory drive the GPe neuron locks when it is entrained and the distribution of phases at which it spikes when it is not. Then, we show that in this case the pairwise spike cross-correlation equals the cross-correlation function of these spike phase distributions. Finally, we show that the distribution of GPe phase delays arises from the diversity of iPRCs and is broadened when the neurons become entrained. Modeling GPe networks with realistic intranuclear connectivity demonstrates that the connectivity decorrelates GPe neurons without affecting phase delays. Thus, common oscillatory input gives rise to GPe correlations whose structure and pairwise phase delays reflect their intrinsic properties captured by their iPRCs.

Propagation of Oscillations in the Indirect Pathway of the Basal Ganglia

Oscillatory signals propagate in the basal ganglia from prototypic neurons in the external globus pallidus (GPe) to their target neurons in the substantia nigra pars reticulata (SNr), internal pallidal segment, and subthalamic nucleus. Neurons in the GPe fire spontaneously, so oscillatory input signals can be encoded as changes in timing of action potentials within an ongoing spike train. When GPe neurons were driven by an oscillatory current in male and female mice, these spike-timing changes produced spike-oscillation coherence over a range of frequencies extending at least to 100 Hz. Using the known kinetics of the GPe→SNr synapse, we calculated the postsynaptic currents that would be generated in SNr neurons from the recorded GPe spike trains. The ongoing synaptic barrage from spontaneous firing, frequency-dependent short-term depression, and stochastic fluctuations at the synapse embed the input oscillation into a noisy sequence of synaptic currents in the SNr. The oscillatory component of the resulting synaptic current must compete with the noisy spontaneous synaptic barrage for control of postsynaptic SNr neurons, which have their own frequency-dependent sensitivities. Despite this, SNr neurons subjected to synaptic conductance changes generated from recorded GPe neuron firing patterns also became coherent with oscillations over a broad range of frequencies. The presynaptic, synaptic, and postsynaptic frequency sensitivities were all dependent on the firing rates of presynaptic and postsynaptic neurons. Firing rate changes, often assumed to be the propagating signal in these circuits, do not encode most oscillation frequencies, but instead determine which signal frequencies propagate effectively and which are suppressed.SIGNIFICANCE STATEMENT Oscillations are present in all the basal ganglia nuclei, include a range of frequencies, and change over the course of learning and behavior. Exaggerated oscillations are a hallmark of basal ganglia pathologies, and each has a specific frequency range. Because of its position as a hub in the basal ganglia circuitry, the globus pallidus is a candidate origin for oscillations propagating between nuclei. We imposed low-amplitude oscillations on individual globus pallidus neurons at specific frequencies and measured the coherence between the oscillation and firing as a function of frequency. We then used these responses to measure the effectiveness of oscillatory propagation to other basal ganglia nuclei. Propagation was effective for oscillation frequencies as high as 100 Hz.

Spontaneous Activity of the Local GABAergic Synaptic Network Causes Irregular Neuronal Firing in the External Globus Pallidus

Autonomously firing GABAergic neurons in the external globus pallidus (GPe) form a local synaptic network. In slices, most GPe neurons receive a continuous inhibitory synaptic barrage from 1 or 2 presynaptic GPe neurons. We measured the barrage's effect on the firing rate and regularity of GPe neurons in male and female mice using perforated patch recordings. Silencing the firing of parvalbumin-positive (PV+) GPe neurons by activating genetically expressed Archaerhodopsin current increased the firing rate and regularity of PV- neurons. In contrast, silencing Npas1+ GPe neurons with Archaerhodopsin had insignificant effects on Npas1- neuron firing. Blocking spontaneous GABAergic synaptic input with gabazine reproduced the effects of silencing PV+ neuron firing on the firing rate and regularity of Npas1+ neurons and had similar effects on PV+ neuron firing. To simulate the barrage, we constructed conductance waveforms for dynamic clamp based on experimentally measured inhibitory postsynaptic conductance trains from 1 or 2 unitary local connections. The resulting inhibition replicated the effect on firing seen in the intact active network in the slice. We then increased the number of unitary inputs to match estimates of local network connectivity in vivo As few as 5 unitary inputs produced large increases in firing irregularity. The firing rate was also reduced initially, but PV+ neurons exhibited a slow spike-frequency adaptation that partially restored the rate despite sustained inhibition. We conclude that the irregular firing pattern of GPe neurons in vivo is largely due to the ongoing local inhibitory synaptic barrage produced by the spontaneous firing of other GPe neurons.SIGNIFICANCE STATEMENT Functional roles of local axon collaterals in the external globus pallidus (GPe) have remained elusive because of difficulty in isolating local inhibition from other GABAergic inputs in vivo, and in preserving the autonomous firing of GPe neurons and detecting their spontaneous local inputs in slices. We used perforated patch recordings to detect spontaneous local inputs during rhythmic firing. We found that the autonomous firing of single presynaptic GPe neurons produces inhibitory synaptic barrages that significantly alter the firing regularity of other GPe neurons. Our findings suggest that, although GPe neurons receive input from only a few other GPe neurons, each local connection has a large impact on their firing.

Local inhibition in a model of the indirect pathway globus pallidus network slows and deregularizes background firing, but sharpens and synchronizes responses to striatal input

The external segment of globus pallidus (GPe) is a network of oscillatory neurons connected by inhibitory synapses. We studied the intrinsic dynamic and the response to a shared brief inhibitory stimulus in a model GPe network. Individual neurons were simulated using a phase resetting model based on measurements from mouse GPe neurons studied in slices. The neurons showed a broad heterogeneity in their firing rates and in the shapes and sizes of their phase resetting curves. Connectivity in the network was set to match experimental measurements. We generated statistically equivalent neuron heterogeneity in a small-world model, in which 99% of connections were made with near neighbors and 1% at random, and in a model with entirely random connectivity. In both networks, the resting activity was slowed and made more irregular by the local inhibition, but it did not show any periodic pattern. Cross-correlations among neuron pairs were limited to directly connected neurons. When stimulated by a shared inhibitory input, the individual neuron responses separated into two groups: one with a short and stereotyped period of inhibition followed by a transient increase in firing probability, and the other responding with a sustained inhibition. Despite differences in firing rate, the responses of the first group of neurons were of fixed duration and were synchronized across cells.

Broadband Entrainment of Striatal Low-Threshold Spike Interneurons

Striatal interneurons and spiny projection (SP) neurons are differentially tuned to spectral components of their input. Previous studies showed that spike responses of somatostatin/NPY-expressing low threshold spike (LTS) interneurons have broad frequency tuning, setting these cells apart from other striatal GABAergic interneurons and SP neurons. We investigated the mechanism of LTS interneuron spiking resonance and its relationship to non-spiking membrane impedance resonance, finding that abolition of impedance resonance did not alter spiking resonance. Because LTS interneurons are pacemakers whose rhythmic firing is perturbed by synaptic input, we tested the hypothesis that their spiking resonance arises from their phase resetting properties. Phase resetting curves (PRCs) were measured in LTS interneurons and SP neurons and used to make phase-oscillator models of both cell types. The models reproduced the broad tuning of LTS interneurons, and the differences from SP neurons. The spectral components of the PRC predicted each cell's sensitivity to corresponding input frequencies. LTS interneuron PRCs contain larger high-frequency components than SP neuron PRCs, providing enhanced responses to input frequencies above the cells' average firing rates. Thus, LTS cells can be entrained by input oscillations to which SP neurons are less responsive. These findings suggest that feedforward inhibition by LTS interneurons may regulate SP neurons' entrainment by oscillatory afferents.

Indirect pathway control of firing rate and pattern in the substantia nigra pars reticulata

Unitary pallido-nigral synaptic currents were measured using optogenetic stimulation, which activated up to three unitary synaptic inputs to each substantia nigra pars reticulata (SNr) cell. Episodic barrages of synaptic conductances were generated based on in vivo firing patterns of globus pallidus pars externa (GPe) cells and applied to SNr cells using conductance clamp. Barrage inputs were compared to continuous step conductances with the same mean. Barrage inputs and steps both slowed SNr neuron firing and produced disinhibition responses seen in peristimulus histograms. Barrages were less effective than steps at producing inhibition and disinhibition responses. Barrages, but not steps, produced irregular firing during the inhibitory response. Phase models of SNr neurons were constructed from their phase-resetting curves. The phase models reproduced the inhibition and disinhibition responses to the same inputs applied to the neurons. The disinhibition response did not require rebound currents but arose from reset of the cells' oscillation. The differences in firing rate and irregularity in response to barrage and step inhibition resulted from the high sensitivity of SNr neurons to inhibition at late phases in their intrinsic oscillation. During step inhibition, cells continued rhythmic firing at a reduced rate. During barrages, brief bouts of intense inhibition stalled the cells' phase evolution late in their cycle, close to firing, and even very brief respites from inhibition rapidly released single action potentials. The SNr cell firing pattern reflected the fine structure of the synaptic barrage from GPe, as well as its onset and offset.NEW & NOTEWORTHY The pallido-nigral pathway connects the striatum to spontaneously active basal ganglia output neurons in the substantia nigra. Each substantia nigra neuron receives powerful inhibitory synaptic connections from a small group of globus pallidus cells and may fire during pauses in pallidal activity. Despite lacking any hyperpolarization-activated rebound currents, they fire quickly to even brief pauses in the pallido-nigral inhibition. The mechanism of their rapid disinhibitory response is explained by features of their phase-resetting curves.

Frequency-dependent entrainment of striatal fast-spiking interneurons

Striatal fast-spiking interneurons (FSIs) fire in variable-length runs of action potentials at 20-200 spikes/s separated by pauses. In vivo, or with fluctuating applied current, both runs and pauses become briefer and more variable. During runs, spikes are entrained specifically to gamma-frequency components of the input fluctuations. We stimulated parvalbumin-expressing striatal FSIs in mouse brain slices with broadband noise currents added to direct current steps and measured spike entrainment across all frequencies. As the constant current level was increased, FSIs produced longer runs and showed sharper frequency tuning, with best entrainment at the stimulus frequency matching their intrarun firing rate. We separated the contributions of previous spikes from that of the fluctuating stimulus, revealing a strong contribution of previous action potentials to gamma-frequency entrainment. In contrast, after subtraction of the effect inherited from the previous spike, the remaining stimulus contribution to spike generation was less sharply tuned, showing a larger contribution of lower frequencies. The frequency specificity of entrainment within a run was reproduced with a phase resetting model based on experimentally measured phase resetting curves of the same FSIs. In the model, broadly tuned phase entrainment for the first spike in a run evolved into sharply tuned gamma entrainment over the next few spikes. The data and modeling results indicate that for FSIs firing in brief runs and pauses firing within runs is entrained by gamma-frequency components of the input, whereas the onset timing of runs may be sensitive to a wider range of stimulus frequency components.NEW & NOTEWORTHY Specific types of neurons entrain their spikes to particular oscillation frequencies in their synaptic input. This entrainment is commonly understood in terms of the subthreshold voltage response, but how this translates to spiking is not clear. We show that in striatal fast-spiking interneurons, entrainment to gamma-frequency input depends on rhythmic spike runs and is explained by the phase resetting curve, whereas run initiation can be triggered by a broad range of input frequencies.

Population dynamics and entrainment of basal ganglia pacemakers are shaped by their dendritic arbors

The theory of phase oscillators is an essential tool for understanding population dynamics of pacemaking neurons. GABAergic pacemakers in the substantia nigra pars reticulata (SNr), a main basal ganglia (BG) output nucleus, receive inputs from the direct and indirect pathways at distal and proximal regions of their dendritic arbors, respectively. We combine theory, optogenetic stimulation and electrophysiological experiments in acute brain slices to ask how dendritic properties impact the propensity of the various inputs, arriving at different locations along the dendrite, to recruit or entrain SNr pacemakers. By combining cable theory with sinusoidally-modulated optogenetic activation of either proximal somatodendritic regions or the entire somatodendritic arbor of SNr neurons, we construct an analytical model that accurately fits the empirically measured somatic current response to inputs arising from illuminating the soma and various portions of the dendritic field. We show that the extent of the dendritic tree that is illuminated generates measurable and systematic differences in the pacemaker’s phase response curve (PRC), causing a shift in its peak. Finally, we show that the divergent PRCs correctly predict differences in two major features of the collective dynamics of SNr neurons: the fidelity of population responses to sudden step-like changes in inputs; and the phase latency at which SNr neurons are entrained by rhythmic stimulation, which can occur in the BG under both physiological and pathophysiological conditions. Our novel method generates measurable and physiologically meaningful spatial effects, and provides the first empirical demonstration of how the collective responses of SNr pacemakers are determined by the transmission properties of their dendrites. SNr dendrites may serve to delay distal striatal inputs so that they impinge on the spike initiation zone simultaneously with pallidal and subthalamic inputs in order to guarantee a fair competition between the influence of the monosynaptic direct- and polysynaptic indirect pathways.

The substantia nigra pars reticulata (SNr) is a main output nucleus of the basal ganglia (BG), where inputs from the competing direct and indirect pathways converge onto the same neurons. Interestingly, these inputs are differentially distributed with direct and indirect pathway projections arriving at distal and proximal regions of the dendritic arbor, respectively. We employ a novel method combining theory with electrophysiological experiments and optogenetics to study the distinct effects of inputs arriving at different locations along the dendrite. Our approach represents a useful compromise between complexity and reduction in modelling. Our work addresses the question of high fidelity encoding of inputs by networks of neurons in the new context of pacemaking neurons, which are driven to fire by their intrinsic dynamics rather than by a network state. We provide the first empirical demonstration that dendritic delays can introduce latencies in the responses of a population of neurons that are commensurate with synaptic delays, suggesting a new role for SNr dendrites with implications for BG function.

Predicting responses to inhibitory synaptic input in substantia nigra pars reticulata neurons

The changes in firing probability produced by a synaptic input are usually visualized using the poststimulus time histogram (PSTH). It would be useful if postsynaptic firing patterns could be predicted from patterns of afferent synaptic activation, but attempts to predict the PSTH from synaptic potential waveforms using reasoning based on voltage trajectory and spike threshold have not been successful, especially for inhibitory inputs. We measured PSTHs for substantia nigra pars reticulata (SNr) neurons inhibited by optogenetic stimulation of striato-nigral inputs or by matching artificial inhibitory conductances applied by dynamic clamp. The PSTH was predicted by a model based on each SNr cell's phase-resetting curve (PRC). Optogenetic activation of striato-nigral input or artificial synaptic inhibition produced a PSTH consisting of an initial depression of firing followed by oscillatory increases and decreases repeating at the SNr cell's baseline firing rate. The phase resetting model produced PSTHs closely resembling the cell data, including the primary pause in firing and the oscillation. Key features of the PSTH, including the onset rate and duration of the initial inhibitory phase, and the subsequent increase in firing probability could be explained from the characteristic shape of the SNr cell's PRC. The rate of damping of the late oscillation was explained by the influence of asynchronous phase perturbations producing firing rate jitter and wander. Our results demonstrate the utility of phase-resetting models as a general method for predicting firing in spontaneously active neurons and their value in interpretation of the striato-nigral PSTH. NEW & NOTEWORTHY The coupling of patterned presynaptic input to sequences of postsynaptic firing is a Gordian knot, complicated by the multidimensionality of neuronal state and the diversity of potential initial states. Even so, it is fundamental for even the simplest understanding of network dynamics. We show that a simple phase-resetting model constructed from experimental measurements can explain and predict the sequence of spike rate changes following synaptic inhibition of an oscillating basal ganglia output neuron.

Measurement of phase resetting curves using optogenetic barrage stimuli

BACKGROUND

The phase resetting curve (PRC) is a primary measure of a rhythmically firing neuron's responses to synaptic input, quantifying the change in phase of the firing oscillation as a function of the input phase. PRCs provide information about whether neurons will synchronize due to synaptic coupling or shared input. However, PRC estimation has been limited to in vitro preparations where stable intracellular recordings can be obtained and background activity is minimal, and new methods are required for in vivo applications.

NEW METHOD

We estimated PRCs using dense optogenetic stimuli and extracellular spike recording. Autonomously firing neurons in substantia nigra pars reticulata (SNr) of Thy1-channelrhodopsin 2 (ChR2) transgenic mice were stimulated with random barrages of light pulses, and PRCs were determined using multiple linear regression.

RESULTS

The PRCs obtained were type-I, showing only phase advances in response to depolarizing input, and generally sloped upward from early to late phases. Secondary PRCs, indicating the effect on the subsequent ISI, showed phase delays primarily for stimuli arriving at late phases. Phase models constructed from the optogenetic PRCs accounted for a large fraction of the variance in ISI length and provided a good approximation of the spike-triggered average stimulus.

COMPARISON WITH EXISTING METHODS

Compared to methods based on intracellular current injection, the new method sacrifices some temporal resolution. However, it should be much more widely applicable in vivo, because only extracellular recording and optogenetic stimulation are required.

CONCLUSIONS

These results demonstrate PRC estimation using methods suitable for in vivo applications.

Hard real-time closed-loop electrophysiology with the Real-Time eXperiment Interface (RTXI)

The ability to experimentally perturb biological systems has traditionally been limited to static pre-programmed or operator-controlled protocols. In contrast, real-time control allows dynamic probing of biological systems with perturbations that are computed on-the-fly during experimentation. Real-time control applications for biological research are available; however, these systems are costly and often restrict the flexibility and customization of experimental protocols. The Real-Time eXperiment Interface (RTXI) is an open source software platform for achieving hard real-time data acquisition and closed-loop control in biological experiments while retaining the flexibility needed for experimental settings. RTXI has enabled users to implement complex custom closed-loop protocols in single cell, cell network, animal, and human electrophysiology studies. RTXI is also used as a free and open source, customizable electrophysiology platform in open-loop studies requiring online data acquisition, processing, and visualization. RTXI is easy to install, can be used with an extensive range of external experimentation and data acquisition hardware, and includes standard modules for implementing common electrophysiology protocols.

Predicting the response of striatal spiny neurons to sinusoidal input

Spike-timing effects of small-amplitude sinusoidal currents were measured in mouse striatal spiny neurons firing repetitively. Spike-timing reliability varied with the stimulus frequency. For frequencies near the cell's firing rate, the cells altered firing rate to match the stimulus and became phase locked to it. The stimulus phase of firing during lock depended on the stimulus frequency relative to the cell's unperturbed firing rate. Interspike intervals during sinusoidal stimulation were predicted using an iterative map constructed from the cells' phase-resetting curve. Variability of interspike intervals was reduced by stimulation at all frequencies higher than about half the cell's unperturbed rate, and interspike intervals were accurately predicted by the map. Long sequences of spike times were predicted by iterating on the map. The accuracy of that prediction varied with frequency. Spike time predictability was highest near and during phase lock. The map predicted the phase of firing on the input and its dependence on stimulus frequency. Prediction errors, when they occurred, were of two kinds: unpredicted variation in interspike interval from intrinsic cell noise and accumulation of prediction errors from previous interspike intervals. Each type of prediction error arose from a different mechanism, and their impact was also predicted from the phase model. When two oscillatory input currents were presented simultaneously, striatal neurons responded selectively to only one of them, the one closest in frequency to the cell's unperturbed firing rate. Their spike times encoded the frequency and phase of that single oscillatory input.NEW & NOTEWORTHY During repetitive firing, the timing of action potentials is determined by the interaction between the input and voltage-sensitive currents throughout the interspike interval. This interaction is encapsulated in the neuron's phase-resetting curve. The phase-resetting curve predicted spike timing to small sinusoidal currents over a wide range of stimulus frequencies. Firing patterns were most sensitive to oscillatory components near the cell's own firing rate, even in the presence of noise and other inputs.

Neurons as oscillators

Regularly spiking neurons can be described as oscillators. In this article we review some of the insights gained from this conceptualization and their relevance for systems neuroscience. First, we explain how a regularly spiking neuron can be viewed as an oscillator and how the phase-response curve (PRC) describes the response of the neuron's spike times to small perturbations. We then discuss the meaning of the PRC for a single neuron's spiking behavior and review the PRCs measured from a variety of neurons in a range of spiking regimes. Next, we show how the PRC can be related to a number of common measures used to quantify neuronal firing, such as the spike-triggered average and the peristimulus histogram. We further show that the response of a neuron to correlated inputs depends on the shape of the PRC. We then explain how the PRC of single neurons can be used to predict neural network behavior. Given the PRC, conduction delays, and the waveform and time course of the synaptic potentials, it is possible to predict neural population behavior such as synchronization. The PRC also allows us to quantify the robustness of the synchronization to heterogeneity and noise. We finally ask how to combine the measured PRCs and the predictions based on PRC to further the understanding of systems neuroscience. As an example, we discuss how the change of the PRC by the neuromodulator acetylcholine could lead to a destabilization of cortical network dynamics. Although all of these studies are grounded in mathematical abstractions that do not strictly hold in biology, they provide good estimates for the emergence of the brain's network activity from the properties of individual neurons. The study of neurons as oscillators can provide testable hypotheses and mechanistic explanations for systems neuroscience.

Unitary synaptic connections among substantia nigra pars reticulata neurons

Neurons in substantia nigra pars reticulata (SNr) are synaptically coupled by local axon collaterals, providing a potential mechanism for local signal processing. Because SNr neurons fire spontaneously, these synapses are constantly active. To investigate their properties, we recorded spontaneous inhibitory postsynaptic currents (sIPSCs) from SNr neurons in brain slices, in which afferents from upstream nuclei are severed, and the cells fire rhythmically. The sIPSC trains contained a mixture of periodic and aperiodic events. Autocorrelation analysis of sIPSC trains showed that a majority of cells had one to four active unitary inputs. The properties of the unitary IPSCs (uIPSCs) were analyzed for cells with one unitary input, using a model of periodic presynaptic firing and stochastic synaptic transmission. The inferred presynaptic firing rates and coefficient of variation of interspike intervals (ISIs) corresponded well with direct measurements of spiking in SNr neurons. Methods were developed to estimate the success probability, amplitude distributions, and kinetics of the uIPSCs, while removing the contribution from aperiodic sIPSCs. The sIPSC amplitudes were not increased upon release from halorhodopsin silencing, suggesting that most synapses were not depressed at the spontaneous firing rate. Gramicidin perforated-patch recordings indicated that the average reversal potential of spontaneous inhibitory postsynaptic potentials was -64 mV. Because of the change in driving force across the ISI, the unitary inputs are predicted to have a larger postsynaptic impact when they arrive late in the ISI. Simulations of network activity suggest that this very sparse inhibitory coupling may act to desynchronize the activity of SNr neurons while having only a small effect on firing rate.

Differential effects of conductances on the phase resetting curve of a bursting neuronal oscillator

The intrinsically oscillating neurons in the crustacean pyloric circuit have membrane conductances that influence their spontaneous activity patterns and responses to synaptic activity. The relationship between the magnitudes of these membrane conductances and the response of the oscillating neurons to synaptic input has not yet been fully or systematically explored. We examined this relationship using the phase resetting curve (PRC), which summarizes the change in the cycle period of a neuronal oscillator as a function of the input's timing within the oscillation. We first utilized a large database of single-compartment model neurons to determine the effect of individual membrane conductances on PRC shape; we found that the effects vary across conductance space, but on average, the hyperpolarization-activated and leak conductances advance the PRC. We next investigated how membrane conductances affect PRCs of the isolated pacemaker kernel in the pyloric circuit of Cancer borealis by: (1) tabulating PRCs while using dynamic clamp to artificially add varying levels of specific conductances, and (2) tabulating PRCs before and after blocking the endogenous hyperpolarization-activated current. We additionally used a previously described four-compartment model to determine how the location of the hyperpolarization-activated conductance influences that current's effect on the PRC. We report that while dynamic-clamp-injected leak current has much smaller effects on the PRC than suggested by the single-compartment model, an increase in the hyperpolarization-activated conductance both advances and reduces the noisiness of the PRC in the pacemaker kernel of the pyloric circuit in both modeling and experimental studies.

On the firing rate dependency of the phase response curve of rat Purkinje neurons in vitro

Synchronous spiking during cerebellar tasks has been observed across Purkinje cells: however, little is known about the intrinsic cellular mechanisms responsible for its initiation, cessation and stability. The Phase Response Curve (PRC), a simple input-output characterization of single cells, can provide insights into individual and collective properties of neurons and networks, by quantifying the impact of an infinitesimal depolarizing current pulse on the time of occurrence of subsequent action potentials, while a neuron is firing tonically. Recently, the PRC theory applied to cerebellar Purkinje cells revealed that these behave as phase-independent integrators at low firing rates, and switch to a phase-dependent mode at high rates. Given the implications for computation and information processing in the cerebellum and the possible role of synchrony in the communication with its post-synaptic targets, we further explored the firing rate dependency of the PRC in Purkinje cells. We isolated key factors for the experimental estimation of the PRC and developed a closed-loop approach to reliably compute the PRC across diverse firing rates in the same cell. Our results show unambiguously that the PRC of individual Purkinje cells is firing rate dependent and that it smoothly transitions from phase independent integrator to a phase dependent mode. Using computational models we show that neither channel noise nor a realistic cell morphology are responsible for the rate dependent shift in the phase response curve.

Phase-resetting as a tool of information transmission

Models of information transmission in the brain largely rely on firing rate codes. The abundance of oscillatory activity in the brain suggests that information may be also encoded using the phases of ongoing oscillations. Sensory perception, working memory and spatial navigation have been hypothesized to use phase codes, and cross-frequency coordination and phase synchronization between brain areas have been proposed to gate the flow of information. Phase codes generally require the phase of the oscillations to be reset at specific reference points for consistent coding, and coordination between oscillators requires favorable phase resetting characteristics. Recent evidence supports a role for neural oscillations in providing temporal reference windows that allow for correct parsing of phase-coded information.

Oscillators and Oscillations in the Basal Ganglia

What is the meaning of an action potential? There must be different answers for neurons that fire spontaneously, even in the absence of synaptic input, and those driven to fire from a resting membrane potential. In spontaneously firing neurons, the occurrence of the next action potential is guaranteed; only variations in its timing can carry the message. In the basal ganglia, the globus pallidus, the substantia nigra, and the subthalamic nucleus consist of neurons firing spontaneously. They each receive thousands of synaptic inputs, but these are not required to maintain their background firing. Instead, synaptic interactions among basal ganglia nuclei comprise a system of coupled oscillators that produces a complex resting pattern of activity. Normally, this pattern is highly irregular and uncorrelated, so that the firing of each cell is statistically independent of the others. This maximizes the potential information that may be transmitted by the basal ganglia to its target structures. In Parkinson's disease, the resting pattern of activity is dominated by a slow oscillation shared by nearly all of the neurons. Treatment with deep brain stimulation may gain its therapeutic value by disrupting this shared pathological oscillation, and restoring independent action by each neuron in the network.