Optimal spectral templates for triggered feedback experiments.

In the field of songbird neuroscience, researchers have used playback of aversive noise bursts to drive changes in song behavior for specific syllables within a bird's song. Typically, a short (~5-10 msec) slice of the syllable is selected for targeting and the average spectrum of the slice is used as a template. Sounds that are sufficiently close to the template are considered a match. If other syllables have portions that are spectrally similar to the target, false positive errors will weaken the operant contingency. We present a gradient descent method for template optimization that increases the separation in distance between target and distractors slices, greatly improving targeting accuracy. Applied to songs from five adult Bengalese finches, the fractional reduction in errors for sub-syllabic slices was 51.54±22.92%. At the level of song syllables, we use an error metric that controls for the vastly greater number of distractors vs. target syllables. Setting 5% average error (misses + false positives) as a minimal performance criterion, the number of targetable syllables increased from 3 to 16 out of 61 syllables. At 10% error, targetable syllables increased from 11 to 26. By using simple and robust linear discriminant methods, the algorithm reaches near asymptotic performance when using 10 songs as training data, and the error increases by <2.3% on average when using only a single song for training. Targeting is temporally precise, with average jitter of 3.33 msec for the 16 accurately targeted syllables. Because the algorithm is concerned only with the problem of template selection, it can be used as a simple and robust front end for existing hardware and software implementations for triggered feedback.

Timing during transitions in Bengalese finch song: implications for motor sequencing.

To investigate mechanisms of action sequencing, we examined the relationship between timing and sequencing of syllables in Bengalese finch song. An individual's song comprises acoustically distinct syllables organized into probabilistic sequences: a given syllable potentially can transition to several different syllables (divergence points), and several different syllables can transition to a given syllable (convergence points). In agreement with previous studies, we found that more probable transitions at divergence points occur with shorter intersyllable gaps. One intuition for this relationship is that selection between syllables reflects a competitive branching process, in which stronger links to one syllable lead to both higher probabilities and shorter latencies for transitions to that syllable vs. competing alternatives. However, we found that simulations of competitive race models result in overlapping winning-time distributions for competing outcomes and fail to replicate the strong negative correlation between probability and gap duration found in song data. Further investigation of song structure revealed strong positive correlation between gap durations for transitions that share a common convergent point. Such transitions are not related by a common competitive process, but instead reflect a common terminal syllable. In contrast to gap durations, transition probabilities were not correlated at convergence points. Together, our data suggest that syllable selection happens early during the gap, with gap timing determined chiefly by the latency to syllable initiation. This may result from a process in which probabilistic sequencing is first stabilized, followed by a shortening of the latency to syllables that are sung more often.NEW & NOTEWORTHY Bengalese finch songs consist of probabilistic sequences of syllables. Previous studies revealed a strong negative correlation between transition probability and the duration of intersyllable gaps. We show here that the negative correlation is inconsistent with previous suggestions that timing at syllable transitions is governed by a race between competing alternatives. Rather, the data suggest that syllable selection happens early during the gap, with gap timing determined chiefly by the latency to syllable initiation.

Continuous Time Representations of Song in Zebra Finches.

Neurons in the songbird nucleus HVC produce premotor bursts time locked to song with millisecond precision. In this issue of Neuron, Lynch et al. (2016) and Picardo et al. (2016) provide convincing evidence that the population of these bursts contain a continuous representation of time throughout song.

Phase locking asymmetries at flexor-extensor transitions during fictive locomotion.

The motor output for walking is produced by a network of neurons termed the spinal central pattern generator (CPG) for locomotion. The basic building block of this CPG is a half-center oscillator composed of two mutually inhibitory sets of interneurons, each controlling one of the two dominant phases of locomotion: flexion and extension. To investigate symmetry between the two components of this oscillator, we analyzed the statistics of natural variation in timing during fictive locomotion induced by stimulation of the midbrain locomotor region in the cat. As a complement to previously published analysis of these data focused on burst and cycle durations, we present a new analysis examining the strength of phase locking at the transitions between flexion and extension. Across our sample of nerve pairs, phase locking at the transition from extension to flexion (E to F) is stronger than at the transition from flexion to extension (F to E). This pattern did not reverse when considering bouts of fictive locomotion that were flexor vs. extensor dominated, demonstrating that asymmetric locking at the transitions between phases is dissociable from which phase dominates cycle duration. We also find that the strength of phase locking is correlated with the mean latency between burst offset and burst onset. These results are interpreted in the context of a hypothesis where network inhibition and intrinsic oscillatory mechanisms make distinct contributions to flexor-extensor alternation in half-center networks.

Development of temporal structure in zebra finch song.

Zebra finch song has provided an excellent case study in the neural basis of sequence learning, with a high degree of temporal precision and tight links with precisely timed bursting in forebrain neurons. To examine the development of song timing, we measured the following four aspects of song temporal structure at four age ranges between 65 and 375 days posthatch: the mean durations of song syllables and the silent gaps between them, timing variability linked to song tempo, timing variability expressed independently across syllables and gaps, and transition probabilities between consecutive syllable pairs. We found substantial increases in song tempo between 65 and 85 days posthatch, due almost entirely to a shortening of gaps. We also found a decrease in tempo variability, also specific to gaps. Both the magnitude of the increase in tempo and the decrease in tempo variability were correlated on gap-by-gap basis with increases in the reliability of corresponding syllable transitions. Syllables had no systematic increase in tempo or decrease in tempo variability. In contrast to tempo parameters, both syllables and gaps showed an early sharp reduction in independent variability followed by continued reductions over the first year. The data suggest that links between syllable-based representations are strengthened during the later parts of the traditional period of song learning and that song rhythm continues to become more regular throughout the first year of life. Similar learning patterns have been identified in human sequence learning, suggesting a potentially rich area of comparative research.

A generative model for measuring latent timing structure in motor sequences.

Motor variability often reflects a mixture of different neural and peripheral sources operating over a range of timescales. We present a statistical model of sequence timing that can be used to measure three distinct components of timing variability: global tempo changes that are spread across the sequence, such as might stem from neuromodulatory sources with widespread influence; fast, uncorrelated timing noise, stemming from noisy components within the neural system; and timing jitter that does not alter the timing of subsequent elements, such as might be caused by variation in the motor periphery or by measurement error. In addition to quantifying the variability contributed by each of these latent factors in the data, the approach assigns maximum likelihood estimates of each factor on a trial-to-trial basis. We applied the model to adult zebra finch song, a temporally complex behavior with rich structure on multiple timescales. We find that individual song vocalizations (syllables) contain roughly equal amounts of variability in each of the three components while overall song length is dominated by global tempo changes. Across our sample of syllables, both global and independent variability scale with average length while timing jitter does not, a pattern consistent with the Wing and Kristofferson (1973) model of sequence timing. We also find significant day-to-day drift in all three timing sources, but a circadian pattern in tempo only. In tests using artificially generated data, the model successfully separates out the different components with small error. The approach provides a general framework for extracting distinct sources of timing variability within action sequences, and can be applied to neural and behavioral data from a wide array of systems.

Disinhibition bursting of dopaminergic neurons.

Substantia nigra pars compacta (SNpc) dopaminergic neurons receive strong tonic inputs from GABAergic neurons in the substantia nigra pars reticulata (SNpr) and globus pallidus (GP), and glutamatergic neurons in the subthalamic nucleus. The presence of these tonic inputs raises the possibility that phasic disinhibition may trigger phasic bursts in dopaminergic neurons. We first applied constant NMDA and GABA(A) conductances onto a two-compartment single cell model of the dopaminergic neuron (Kuznetsov et al., 2006). The model exhibited disinhibition bursting upon stepwise removal of inhibition. A further bifurcation analysis suggests that disinhibition may be more robust than excitation alone in that for most levels of NMDA conductance, the cell remains capable of bursting even after a complete removal of inhibition, whereas too much excitatory input will drive the cell into depolarization block. To investigate the network dynamics of disinhibition, we used a modified version of an integrate-and-fire based model of the basal ganglia (Humphries et al., 2006). Synaptic activity generated in the network was delivered to the two-compartment single cell dopaminergic neuron. Phasic activation of the D1-expressing medium spiny neurons in the striatum (D1STR) produced disinhibition bursts in dopaminergic neurons through the direct pathway (D1STR to SNpr to SNpc). Anatomical studies have shown that D1STR neurons have collaterals that terminate in GP. Adding these collaterals to the model, we found that striatal activation increased the intra-burst firing frequency of the disinhibition burst as the weight of this connection was increased. Our studies suggest that striatal activation is a robust means by which disinhibition bursts can be generated by SNpc dopaminergic neurons, and that recruitment of the indirect pathway via collaterals may enhance disinhibition bursting.

The dynamics of integrate-and-fire: mean versus variance modulations and dependence on baseline parameters.

The leaky integrate-and-fire (LIF) is the simplest neuron model that captures the essential properties of neuronal signaling. Yet common intuitions are inadequate to explain basic properties of LIF responses to sinusoidal modulations of the input. Here we examine responses to low and moderate frequency modulations of both the mean and variance of the input current and quantify how these responses depend on baseline parameters. Across parameters, responses to modulations in the mean current are low pass, approaching zero in the limit of high frequencies. For very low baseline firing rates, the response cutoff frequency matches that expected from membrane integration. However, the cutoff shows a rapid, supralinear increase with firing rate, with a steeper increase in the case of lower noise. For modulations of the input variance, the gain at high frequency remains finite. Here, we show that the low-frequency responses depend strongly on baseline parameters and derive an analytic condition specifying the parameters at which responses switch from being dominated by low versus high frequencies. Additionally, we show that the resonant responses for variance modulations have properties not expected for common oscillatory resonances: they peak at frequencies higher than the baseline firing rate and persist when oscillatory spiking is disrupted by high noise. Finally, the responses to mean and variance modulations are shown to have a complementary dependence on baseline parameters at higher frequencies, resulting in responses to modulations of Poisson input rates that are independent of baseline input statistics.

Complementary responses to mean and variance modulations in the perfect integrate-and-fire model.

In the perfect integrate-and-fire model (PIF), the membrane voltage is proportional to the integral of the input current since the time of the previous spike. It has been shown that the firing rate within a noise free ensemble of PIF neurons responds instantaneously to dynamic changes in the input current, whereas in the presence of white noise, model neurons preferentially pass low frequency modulations of the mean current. Here, we prove that when the input variance is perturbed while holding the mean current constant, the PIF responds preferentially to high frequency modulations. Moreover, the linear filters for mean and variance modulations are complementary, adding exactly to one. Since changes in the rate of Poisson distributed inputs lead to proportional changes in the mean and variance, these results imply that an ensemble of PIF neurons transmits a perfect replica of the time-varying input rate for Poisson distributed input. A more general argument shows that this property holds for any signal leading to proportional changes in the mean and variance of the input current.

Behavioral measurements of a temporally precise motor code for birdsong.

There are conflicting data on the timescale for the representation of adult zebra finch song. Acoustic structure and perturbation studies suggest that song is divided into discrete vocal elements, or syllables, lasting 50-200 ms. However, recordings in premotor telencephalic nucleus HVC (used as proper name) and RA (robust nucleus of arcopallium) suggest that song is represented by sparse, fine-grained bursting on the 5-10 ms timescale. We previously found patterns of timing variability that distinguish individual syllables and repeat across multiple 500- to 1000-ms-long motifs (Glaze and Troyer, 2006). Here, we extend our methods to analyze whether this is attributable to a syllable-based code or representations on a finer timescale. We find evidence for the latter. First, identity-dependent timing is dominated by independent variability in notes, finer song segments that compose a syllable; for example, the length of a note is no more correlated with other notes in the same syllable than it is with notes in other syllables. For a subset of notes, clear modulation in spectral structure allowed for accurate timing measurements on the 5-10 ms timescale. Temporal independence holds at this scale as well: the length of an individual 5-10 ms song slice is correlated with the same slice repeated 500-1000 ms later, yet is independent of neighboring slices. We propose that such fine-grained, persistent changes in song tempo result from an interaction between slow modulatory factors and precisely timed, sparse bursting in HVC and RA.

Factors affecting phase synchronization in integrate-and-fire oscillators.

Step changes in input current are known to induce partial phase synchrony in ensembles of leaky integrate-and-fire neurons operating in the oscillatory or "regular firing" regime. An analysis of this phenomenon in the absence of noise is presented based on the probability flux within an ensemble of generalized integrate-and-fire neurons. It is shown that the induction of phase synchrony by a step input can be determined by calculating the ratio of the voltage densities obtained from fully desynchronized ensembles firing at the pre and post-step firing rates. In the limit of low noise and in the absence of phase synchrony, the probability density as a function of voltage is inversely proportional to the time derivative along the voltage trajectory. It follows that the magnitude of phase synchronization depends on the degree to which a change in input leads to a uniform multiplication of the voltage derivative over the range from reset to spike threshold. This analysis is used to investigate several factors affecting phase synchronization including high firing rates, inputs modeled as conductances rather than currents, peri-threshold sodium currents, and spike-triggered potassium currents. Finally, we show that without noise, the equilibrium ensemble density is proportional to the phase response curve commonly used to analyze oscillatory systems.

Temporal structure in zebra finch song: implications for motor coding.

Adult zebra finch songs consist of stereotyped sequences of syllables. Although some behavioral and physiological data suggest that songs are structured hierarchically, there is also evidence that they are driven by nonhierarchical, clock-like bursting in the premotor nucleus HVC (used as a proper name). In this study, we developed a semiautomated template-matching algorithm to identify repeated sequences of syllables and a modified dynamic time-warping algorithm to make fine-grained measurements of the temporal structure of song. We find that changes in song length are expressed across the song as a whole rather than resulting from an accumulation of independent variance during singing. Song length changes systematically over the course of a day and is related to the general level of bird activity as well as the presence of a female. The data also show patterns of variability that suggest distinct mechanisms underlying syllable and gap lengths: as tempo varies, syllables stretch and compress proportionally less than gaps, whereas syllable-syllable and gap-gap correlations are significantly stronger than syllable-gap correlations. There is also increased temporal variability at motif boundaries and especially strong positive correlations between the same syllables sung in different motifs. Finally, we find evidence that syllable onsets may have a special role in aligning syllables with global song structure. Generally, the timing data support a hierarchical view in which song is composed of smaller syllable-based units and provide a rich set of constraints for interpreting the results of physiological recordings.

Temporal correlations in stochastic models of double bursting during simulated locomotion.

The output of the spinal central pattern generator for locomotion falls into two broad categories: alternation between antagonistic muscles and double bursting within muscles acting on multiple joints. We first model an alternating half-center and then present two different models of double bursting. The first double-bursting model consists of a central clock with an explicit one-to-one mapping between interneuron activity and model output. The second double-bursting model consists of a half-center with an added feedback neuron. Models are built using rate-coded leaky integrator neurons with slow self-inhibition. Structure-function relationships are explored by the addition of noise. The interaction of noise with the dynamics of each network creates a unique pattern of correlation between phases of the simulated cycle. The effects of noise can be explained by perturbation of deterministic versions of the networks. Three basic results were obtained: slow self-inhibitory currents lead to correlations between parts of the step cycle that are separated in time and network relative; model outputs are most sensitive to perturbations presented just before competitive switches in network activity, and clock-like models possess substantial symmetries within the correlation structure of burst durations, whereas the correlation structure of feedback models are asymmetric. Our models suggest that variability in burst length durations can be analyzed to make inferences about the structure of the spinal networks for locomotion. In particular, correlation patterns within double-bursting outputs may yield important clues regarding the interaction between more central, clock-like networks and feedback from more peripheral interneurons.

LGN input to simple cells and contrast-invariant orientation tuning: an analysis.

We develop a new analysis of the lateral geniculate nucleus (LGN) input to a cortical simple cell, demonstrating that this input is the sum of two terms, a linear term and a nonlinear term. In response to a drifting grating, the linear term represents the temporal modulation of input, and the nonlinear term represents the mean input. The nonlinear term, which grows with stimulus contrast, has been neglected in many previous models of simple cell response. We then analyze two scenarios by which contrast-invariance of orientation tuning may arise. In the first scenario, at larger contrasts, the nonlinear part of the LGN input, in combination with strong push-pull inhibition, counteracts the nonlinear effects of cortical spike threshold, giving the result that orientation tuning scales with contrast. In the second scenario, at low contrasts, the nonlinear component of LGN input is negligible, and noise smooths the nonlinearity of spike threshold so that the input-output function approximates a power-law function. These scenarios can be combined to yield contrast-invariant tuning over the full range of stimulus contrast. The model clarifies the contribution of LGN nonlinearities to the orientation tuning of simple cells and demonstrates how these nonlinearities may impact different models of contrast-invariant tuning.

Neural noise can explain expansive, power-law nonlinearities in neural response functions.

Many phenomenological models of the responses of simple cells in primary visual cortex have concluded that a cell's firing rate should be given by its input raised to a power greater than one. This is known as an expansive power-law nonlinearity. However, intracellular recordings have shown that a different nonlinearity, a linear-threshold function, appears to give a good prediction of firing rate from a cell's low-pass-filtered voltage response. Using a model based on a linear-threshold function, Anderson et al. showed that voltage noise was critical to converting voltage responses with contrast-invariant orientation tuning into spiking responses with contrast-invariant tuning. We present two separate results clarifying the connection between noise-smoothed linear-threshold functions and power-law nonlinearities. First, we prove analytically that a power-law nonlinearity is the only input-output function that converts contrast-invariant input tuning into contrast-invariant spike tuning. Second, we examine simulations of a simple model that assumes instantaneous spike rate is given by a linear-threshold function of voltage and voltage responses include significant noise. We show that the resulting average spike rate is well described by an expansive power law of the average voltage (averaged over multiple trials), provided that average voltage remains less than about 1.5 SDs of the noise above threshold. Finally, we use this model to show that the noise levels recorded by Anderson et al. are consistent with the degree to which the orientation tuning of spiking responses is more sharply tuned relative to the orientation tuning of voltage responses. Thus neuronal noise can robustly generate power-law input-output functions of the form frequently postulated for simple cells.