Biophysical basis of the phase response curve of subthalamic neurons with generalization to other cell types

The voltage and spiking responses of subthreshold resonant neurons to structured and fluctuating inputs: persistence and loss of resonance and variability

We systematically investigate the response of neurons to oscillatory currents and synaptic-like inputs and we extend our investigation to non-structured synaptic-like spiking inputs with more realistic distributions of presynaptic spike times. We use two types of chirp-like inputs consisting of (i) a sequence of cycles with discretely increasing frequencies over time, and (ii) a sequence having the same cycles arranged in an arbitrary order. We develop and use a number of frequency-dependent voltage response metrics to capture the different aspects of the voltage response, including the standard impedance (Z) and the peak-to-trough amplitude envelope ([Formula: see text]) profiles. We show that Z-resonant cells (cells that exhibit subthreshold resonance in response to sinusoidal inputs) also show [Formula: see text]-resonance in response to sinusoidal inputs, but generally do not (or do it very mildly) in response to square-wave and synaptic-like inputs. In the latter cases the resonant response using Z is not predictive of the preferred frequencies at which the neurons spike when the input amplitude is increased above subthreshold levels. We also show that responses to conductance-based synaptic-like inputs are attenuated as compared to the response to current-based synaptic-like inputs, thus providing an explanation to previous experimental results. These response patterns were strongly dependent on the intrinsic properties of the participating neurons, in particular whether the unperturbed Z-resonant cells had a stable node or a focus. In addition, we show that variability emerges in response to chirp-like inputs with arbitrarily ordered patterns where all signals (trials) in a given protocol have the same frequency content and the only source of uncertainty is the subset of all possible permutations of cycles chosen for a given protocol. This variability is the result of the multiple different ways in which the autonomous transient dynamics is activated across cycles in each signal (different cycle orderings) and across trials. We extend our results to include high-rate Poisson distributed current- and conductance-based synaptic inputs and compare them with similar results using additive Gaussian white noise. We show that the responses to both Poisson-distributed synaptic inputs are attenuated with respect to the responses to Gaussian white noise. For cells that exhibit oscillatory responses to Gaussian white noise (band-pass filters), the response to conductance-based synaptic inputs are low-pass filters, while the response to current-based synaptic inputs may remain band-pass filters, consistent with experimental findings. Our results shed light on the mechanisms of communication of oscillatory activity among neurons in a network via subthreshold oscillations and resonance and the generation of network resonance.

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.

Learning from Action: Reconsidering Movement Signaling in Midbrain Dopamine Neuron Activity

Animals infer when and where a reward is available from experience with informative sensory stimuli and their own actions. In vertebrates, this is thought to depend upon the release of dopamine from midbrain dopaminergic neurons. Studies of the role of dopamine have focused almost exclusively on their encoding of informative sensory stimuli; however, many dopaminergic neurons are active just prior to movement initiation, even in the absence of sensory stimuli. How should current frameworks for understanding the role of dopamine incorporate these observations? To address this question, we review recent anatomical and functional evidence for action-related dopamine signaling. We conclude by proposing a framework in which dopaminergic neurons encode subjective signals of action initiation to solve an internal credit assignment problem.

How to correctly quantify neuronal phase-response curves from noisy recordings

At the level of individual neurons, various coding properties can be inferred from the input-output relationship of a cell. For small inputs, this relation is captured by the phase-response curve (PRC), which measures the effect of a small perturbation on the timing of the subsequent spike. Experimentally, however, an accurate experimental estimation of PRCs is challenging. Despite elaborate measurement efforts, experimental PRC estimates often cannot be related to those from modeling studies. In particular, experimental PRCs rarely resemble the characteristic theoretical PRC expected close to spike initiation, which is indicative of the underlying spike-onset bifurcation. Here, we show for conductance-based model neurons that the correspondence between theoretical and measured phase-response curve is lost when the stimuli used for the estimation are too large. In this case, the derived phase-response curve is distorted beyond recognition and takes on a generic shape that reflects the measurement protocol and masks the spike-onset bifurcation. We discuss how to identify appropriate stimulus strengths for perturbation and noise-stimulation methods, which permit to estimate PRCs that reliably reflect the spike-onset bifurcation - a task that is particularly difficult if a lower bound for the stimulus amplitude is dictated by prominent intrinsic neuronal noise.

Phase-Dependent Suppression of Beta Oscillations in Parkinson's Disease Patients

Synchronized oscillations within and between brain areas facilitate normal processing, but are often amplified in disease. A prominent example is the abnormally sustained beta-frequency (∼20 Hz) oscillations recorded from the cortex and subthalamic nucleus of Parkinson's disease patients. Computational modeling suggests that the amplitude of such oscillations could be modulated by applying stimulation at a specific phase. Such a strategy would allow selective targeting of the oscillation, with relatively little effect on other activity parameters. Here, activity was recorded from 10 awake, parkinsonian patients (6 male, 4 female human subjects) undergoing functional neurosurgery. We demonstrate that stimulation arriving on a particular patient-specific phase of the beta oscillation over consecutive cycles could suppress the amplitude of this pathophysiological activity by up to 40%, while amplification effects were relatively weak. Suppressive effects were accompanied by a reduction in the rhythmic output of subthalamic nucleus (STN) neurons and synchronization with the mesial cortex. While stimulation could alter the spiking pattern of STN neurons, there was no net effect on firing rate, suggesting that reduced beta synchrony was a result of alterations to the relative timing of spiking activity, rather than an overall change in excitability. Together, these results identify a novel intrinsic property of cortico-basal ganglia synchrony that suggests the phase of ongoing neural oscillations could be a viable and effective control signal for the treatment of Parkinson's disease. This work has potential implications for other brain diseases with exaggerated neuronal synchronization and for probing the function of rhythmic activity in the healthy brain.SIGNIFICANCE STATEMENT In Parkinson's disease (PD), movement impairment is correlated with exaggerated beta frequency oscillations in the cerebral cortex and subthalamic nucleus (STN). Using a novel method of stimulation in PD patients undergoing neurosurgery, we demonstrate that STN beta oscillations can be suppressed when consecutive electrical pulses arrive at a specific phase of the oscillation. This effect is likely because of interrupting the timing of neuronal activity rather than excitability, as stimulation altered the firing pattern of STN spiking without changing overall rate. These findings show the potential of oscillation phase as an input for "closed-loop" stimulation, which could provide a valuable neuromodulation strategy for the treatment of brain disorders and for elucidating the role of neuronal oscillations in the healthy brain.

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.

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.

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.

Predicting the responses of repetitively firing neurons to current noise

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.

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.

Spectral reconstruction of phase response curves reveals the synchronization properties of mouse globus pallidus neurons

The propensity of a neuron to synchronize is captured by its infinitesimal phase response curve (iPRC). Determining whether an iPRC is biphasic, meaning that small depolarizing perturbations can actually delay the next spike, if delivered at appropriate phases, is a daunting experimental task because negative lobes in the iPRC (unlike positive ones) tend to be small and may be occluded by the normal discharge variability of a neuron. To circumvent this problem, iPRCs are commonly derived from numerical models of neurons. Here, we propose a novel and natural method to estimate the iPRC by direct estimation of its spectral modes. First, we show analytically that the spectral modes of the iPRC of an arbitrary oscillator are readily measured by applying weak harmonic perturbations. Next, applying this methodology to biophysical neuronal models, we show that a low-dimensional spectral reconstruction is sufficient to capture the structure of the iPRC. This structure was preserved reasonably well even with added physiological scale jitter in the neuronal models. To validate the methodology empirically, we applied it first to a low-noise electronic oscillator with a known design and then to cortical pyramidal neurons, recorded in whole cell configuration, that are known to possess a monophasic iPRC. Finally, using the methodology in conjunction with perforated-patch recordings from pallidal neurons, we show, in contrast to recent modeling studies, that these neurons have biphasic somatic iPRCs. Biphasic iPRCs would cause lateral somatically targeted pallidal inhibition to desynchronize pallidal neurons, providing a plausible explanation for their lack of synchrony in vivo.

Active decorrelation in the basal ganglia

The cytoarchitecturally-homogeneous appearance of the globus pallidus, subthalamic nucleus and substantia nigra has long been said to imply a high degree of afferent convergence and sharing of inputs by nearby neurons. Moreover, axon collaterals of neurons in the external segment of the globus pallidus and the substantia nigra pars reticulata arborize locally and make inhibitory synapses on other cells of the same type. These features suggest that the connectivity of the basal ganglia may impose spike-time correlations among the cells, and it has been puzzling that experimental studies have failed to demonstrate such correlations. One possible solution arises from studies of firing patterns in basal ganglia cells, which reveal that they are nearly all pacemaker cells. Their high rate of firing does not depend on synaptic excitation, but they fire irregularly because a dense barrage of synaptic inputs normally perturbs the timing of their autonomous activity. Theoretical and computational studies show that the responses of repetitively-firing neurons to shared input or mutual synaptic coupling often defy classical intuitions about temporal synaptic integration. The patterns of spike-timing among such neurons depend on the ionic mechanism of pacemaking, the level of background uncorrelated cellular and synaptic noise, and the firing rates of the neurons, as well as the properties of their synaptic connections. Application of these concepts to the basal ganglia circuitry suggests that the connectivity and physiology of these nuclei may be configured to prevent the establishment of permanent spike-timing relationships between neurons. The development of highly synchronous oscillatory patterns of activity in Parkinson's disease may result from the loss of pacemaking by some basal ganglia neurons, and accompanying breakdown of the mechanisms responsible for active decorrelation.

Optimal number of stimulation contacts for coordinated reset neuromodulation

In this computational study we investigate coordinated reset (CR) neuromodulation designed for an effective control of synchronization by multi-site stimulation of neuronal target populations. This method was suggested to effectively counteract pathological neuronal synchrony characteristic for several neurological disorders. We study how many stimulation sites are required for optimal CR-induced desynchronization. We found that a moderate increase of the number of stimulation sites may significantly prolong the post-stimulation desynchronized transient after the stimulation is completely switched off. This can, in turn, reduce the amount of the administered stimulation current for the intermittent ON-OFF CR stimulation protocol, where time intervals with stimulation ON are recurrently followed by time intervals with stimulation OFF. In addition, we found that the optimal number of stimulation sites essentially depends on how strongly the administered current decays within the neuronal tissue with increasing distance from the stimulation site. In particular, for a broad spatial stimulation profile, i.e., for a weak spatial decay rate of the stimulation current, CR stimulation can optimally be delivered via a small number of stimulation sites. Our findings may contribute to an optimization of therapeutic applications of CR neuromodulation.

Effect of sharp jumps at the edges of phase response curves on synchronization of electrically coupled neuronal oscillators

We study synchronization phenomenon of coupled neuronal oscillators using the theory of weakly coupled oscillators. The role of sudden jumps in the phase response curve profiles found in some experimental recordings and models on the ability of coupled neurons to exhibit synchronous and antisynchronous behavior is investigated, when the coupling between the neurons is electrical. The level of jumps in the phase response curve at either end, spike width and frequency of voltage time course of the coupled neurons are parameterized using piecewise linear functional forms, and the conditions for stable synchrony and stable antisynchrony in terms of those parameters are computed analytically. The role of the peak position of the phase response curve on phase-locking is also investigated.

Phase response curves of subthalamic neurons measured with synaptic input and current injection

Infinitesimal phase response curves (iPRCs) provide a simple description of the response of repetitively firing neurons and may be used to predict responses to any pattern of synaptic input. Their simplicity makes them useful for understanding the dynamics of neurons when certain conditions are met. For example, the sizes of evoked phase shifts should scale linearly with stimulus strength, and the form of the iPRC should remain relatively constant as firing rate varies. We measured the PRCs of rat subthalamic neurons in brain slices using corticosubthalamic excitatory postsynaptic potentials (EPSPs; mediated by both AMPA- and NMDA-type receptors) and injected current pulses and used them to calculate the iPRC. These were relatively insensitive to both the size of the stimulus and the cell's firing rate, suggesting that the iPRC can predict the response of subthalamic nucleus cells to extrinsic inputs. However, the iPRC calculated using EPSPs differed from that obtained using current pulses. EPSPs (normalized for charge) were much more effective at altering the phase of subthalamic neurons than current pulses. The difference was not attributable to the extended time course of NMDA receptor-mediated currents, being unaffected by blockade of NMDA receptors. The iPRC provides a good description of subthalamic neurons' response to input, but iPRCs are best estimated using synaptic inputs rather than somatic current injection.