Inferring Neuronal Dynamics from Calcium Imaging Data Using Biophysical Models and Bayesian Inference

Calcium imaging has been used as a promising technique to monitor the dynamic activity of neuronal populations. However, the calcium trace is temporally smeared which restricts the extraction of quantities of interest such as spike trains of individual neurons. To address this issue, spike reconstruction algorithms have been introduced. One limitation of such reconstructions is that the underlying models are not informed about the biophysics of spike and burst generations. Such existing prior knowledge might be useful for constraining the possible solutions of spikes. Here we describe, in a novel Bayesian approach, how principled knowledge about neuronal dynamics can be employed to infer biophysical variables and parameters from fluorescence traces. By using both synthetic and in vitro recorded fluorescence traces, we demonstrate that the new approach is able to reconstruct different repetitive spiking and/or bursting patterns with accurate single spike resolution. Furthermore, we show that the high inference precision of the new approach is preserved even if the fluorescence trace is rather noisy or if the fluorescence transients show slow rise kinetics lasting several hundred milliseconds, and inhomogeneous rise and decay times. In addition, we discuss the use of the new approach for inferring parameter changes, e.g. due to a pharmacological intervention, as well as for inferring complex characteristics of immature neuronal circuits.     

Retention of lung distension information in pump cell spike trains

Respiratory control requires feedback signals from the viscera, including mechanoreceptors and chemoreceptors. We previously showed that typical pulmonary stretch receptor (PSR) spike trains provide the central nervous system with approximately 31% of the theoretical maximum information regarding the amplitude of lung distension. However, it is unknown whether the spatiotemporal convergence of many PSR inputs onto second-order neurons (e.g., pump cells) results in more, or less, information about the stimulus carried by second-order cell spike trains. We recorded pump cell activity in adult, anesthetized, paralyzed, artificially ventilated rabbits during continuous manipulation of ventilator rate and volume to test the hypothesis that less information is carried by spike trains of individual pump cells than PSRs. Using previously developed analytic methods, we quantified the information carried by the pump cell spike trains and compared it with the same values derived from PSR data. Our results provide evidence that rejects our hypothesis: pump cells as a group did not carry significantly less information about the lung distension stimulus than PSRs, although that trend was implied by the data. By comparing the response variances with the theoretical minimum, we discovered that the trend toward information loss depends on response strength, with higher mean responses associated with larger response variances in pump cells than in PSRs. Thus spatiotemporal integration may result in information loss within certain analytic/stimulus parameters, but this is counterbalanced by the consistency of pump cell responses during brief integration times and/or low stimulus amplitudes, resulting in retention of total information.     

A minimal model for decoding of time-limited Ca2+ oscillations

Calcium oscillations regulate several cellular processes by activating particular proteins. Most theoretical studies focused on the idealized situation of infinitely long oscillations. Here we analyze information transfer by time-limited calcium spike trains. We show that proteins can be selectively activated in a resonance-like manner by time-limited spike trains of different frequencies, while infinitely long oscillations do not show this resonance phenomenon. We found that proteins are activated more specifically by shorter oscillatory signals with narrower spikes.     

Estimating Extracellular Spike Waveforms from CA1 Pyramidal Cells with Multichannel Electrodes

Extracellular (EC) recordings of action potentials from the intact brain are embedded in background voltage fluctuations known as the “local field potential” (LFP). In order to use EC spike recordings for studying biophysical properties of neurons, the spike waveforms must be separated from the LFP. Linear low-pass and high-pass filters are usually insufficient to separate spike waveforms from LFP, because they have overlapping frequency bands. Broad-band recordings of LFP and spikes were obtained with a 16-channel laminar electrode array (silicone probe). We developed an algorithm whereby local LFP signals from spike-containing channel were modeled using locally weighted polynomial regression analysis of adjoining channels without spikes. The modeled LFP signal was subtracted from the recording to estimate the embedded spike waveforms. We tested the method both on defined spike waveforms added to LFP recordings, and on in vivo-recorded extracellular spikes from hippocampal CA1 pyramidal cells in anaesthetized mice. We show that the algorithm can correctly extract the spike waveforms embedded in the LFP. In contrast, traditional high-pass filters failed to recover correct spike shapes, albeit produceing smaller standard errors. We found that high-pass RC or 2-pole Butterworth filters with cut-off frequencies below 12.5 Hz, are required to retrieve waveforms comparable to our method. The method was also compared to spike-triggered averages of the broad-band signal, and yielded waveforms with smaller standard errors and less distortion before and after the spike.     

Influence of the temporal distribution of electric pulses on transcallosal single unit responses

We examined how differently timed stimuli to one auditory cortex affect the spike trains they drive in the controlateral homotopic field of anesthetized rats. Bipolar electrical stimulations consisted of trains of pulses (100 micro s, <500 micro A) at rates of 25, 50 or 125 pulses/s and with different stimulus patterns (i.e., dispersions, sequences), called "pacemaker", "accelerando" or "decelerando". Trains lasted for 342 ms and were separated by 4 s. When trains were evaluated over times comparable to the stimulus duration changes characteristically involved an initial slowing followed by recovery and several discharges both stimulus- and neuron-dependent. When evaluated by cross-correlations between cortical cell pairs, the changes extended far beyond the stimulus end. Results suggest that interhemispheric projections, by way of their averages and patterns, play key, long duration roles in the spike-dependent properties of cortical synapses (e.g., potentiation, depression) and thus of cortical circuit operations.     

A Neuron-Based Screening Platform for Optimizing Genetically-Encoded Calcium Indicators

Fluorescent protein-based sensors for detecting neuronal activity have been developed largely based on non-neuronal screening systems. However, the dynamics of neuronal state variables (e.g., voltage, calcium, etc.) are typically very rapid compared to those of non-excitable cells. We developed an electrical stimulation and fluorescence imaging platform based on dissociated rat primary neuronal cultures. We describe its use in testing genetically-encoded calcium indicators (GECIs). Efficient neuronal GECI expression was achieved using lentiviruses containing a neuronal-selective gene promoter. Action potentials (APs) and thus neuronal calcium levels were quantitatively controlled by electrical field stimulation, and fluorescence images were recorded. Images were segmented to extract fluorescence signals corresponding to individual GECI-expressing neurons, which improved sensitivity over full-field measurements. We demonstrate the superiority of screening GECIs in neurons compared with solution measurements. Neuronal screening was useful for efficient identification of variants with both improved response kinetics and high signal amplitudes. This platform can be used to screen many types of sensors with cellular resolution under realistic conditions where neuronal state variables are in relevant ranges with respect to timing and amplitude.     

Influence of different envelope maskers on signal recognition and neuronal representation in the auditory system of a grasshopper

Background

Animals that communicate by sound face the problem that the signals arriving at the receiver often are degraded and masked by noise. Frequency filters in the receiver''s auditory system may improve the signal-to-noise ratio (SNR) by excluding parts of the spectrum which are not occupied by the species-specific signals. This solution, however, is hardly amenable to species that produce broad band signals or have ears with broad frequency tuning. In mammals auditory filters exist that work in the temporal domain of amplitude modulations (AM). Do insects also use this type of filtering?

Principal Findings

Combining behavioural and neurophysiological experiments we investigated whether AM filters may improve the recognition of masked communication signals in grasshoppers. The AM pattern of the sound, its envelope, is crucial for signal recognition in these animals. We degraded the species-specific song by adding random fluctuations to its envelope. Six noise bands were used that differed in their overlap with the spectral content of the song envelope. If AM filters contribute to reduced masking, signal recognition should depend on the degree of overlap between the song envelope spectrum and the noise spectra. Contrary to this prediction, the resistance against signal degradation was the same for five of six masker bands. Most remarkably, the band with the strongest frequency overlap to the natural song envelope (0–100 Hz) impaired acceptance of degraded signals the least. To assess the noise filter capacities of single auditory neurons, the changes of spike trains as a function of the masking level were assessed. Increasing levels of signal degradation in different frequency bands led to similar changes in the spike trains in most neurones.

Conclusions

There is no indication that auditory neurones of grasshoppers are specialized to improve the SNR with respect to the pattern of amplitude modulations.     

Biophysical model of synaptic plasticity dynamics

We discuss a biophysical model of synaptic plasticity that provides a unified view of the outcomes of synaptic modification protocols, including: (1) prescribed time courses of postsynaptic intracellular Ca²⁺ release, (2) postsynaptic voltage clamping with presentation of presynaptic spike trains at various frequencies, (3) direct postsynaptic response to presynaptic spike trains at various frequencies, and (4) LTP/LTD as a response to precisely timed presynaptic and postsynaptic spikes.     

Complementary encoding of spatial information in hippocampal astrocytes

Calcium dynamics into astrocytes influence the activity of nearby neuronal structures. However, because previous reports show that astrocytic calcium signals largely mirror neighboring neuronal activity, current information coding models neglect astrocytes. Using simultaneous two-photon calcium imaging of astrocytes and neurons in the hippocampus of mice navigating a virtual environment, we demonstrate that astrocytic calcium signals encode (i.e., statistically reflect) spatial information that could not be explained by visual cue information. Calcium events carrying spatial information occurred in topographically organized astrocytic subregions. Importantly, astrocytes encoded spatial information that was complementary and synergistic to that carried by neurons, improving spatial position decoding when astrocytic signals were considered alongside neuronal ones. These results suggest that the complementary place dependence of localized astrocytic calcium signals may regulate clusters of nearby synapses, enabling dynamic, context-dependent variations in population coding within brain circuits.

A combination of functional imaging of astrocytes and neurons in the mouse hippocampus with information theory analysis shows that calcium dynamics in topographically-organized subcellular regions of astrocytes encode information about an animal’s position that is complementary and synergistic to that encoded in the spike output of surrounding neurons.     

Fast spatiotemporal smoothing of calcium measurements in dendritic trees

We discuss methods for fast spatiotemporal smoothing of calcium signals in dendritic trees, given single-trial, spatially localized imaging data obtained via multi-photon microscopy. By analyzing the dynamics of calcium binding to probe molecules and the effects of the imaging procedure, we show that calcium concentration can be estimated up to an affine transformation, i.e., an additive and multiplicative constant. To obtain a full spatiotemporal estimate, we model calcium dynamics within the cell using a functional approach. The evolution of calcium concentration is represented through a smaller set of hidden variables that incorporate fast transients due to backpropagating action potentials (bAPs), or other forms of stimulation. Because of the resulting state space structure, inference can be done in linear time using forward-backward maximum-a-posteriori methods. Non-negativity constraints on the calcium concentration can also be incorporated using a log-barrier method that does not affect the computational scaling. Moreover, by exploiting the neuronal tree structure we show that the cost of the algorithm is also linear in the size of the dendritic tree, making the approach applicable to arbitrarily large trees. We apply this algorithm to data obtained from hippocampal CA1 pyramidal cells with experimentally evoked bAPs, some of which were paired with excitatory postsynaptic potentials (EPSPs). The algorithm recovers the timing of the bAPs and provides an estimate of the induced calcium transient throughout the tree. The proposed methods could be used to further understand the interplay between bAPs and EPSPs in synaptic strength modification. More generally, this approach allows us to infer the concentration on intracellular calcium across the dendritic tree from noisy observations at a discrete set of points in space.     

Spectral analysis of input spike trains by spike-timing-dependent plasticity

Spike-timing-dependent plasticity (STDP) has been observed in many brain areas such as sensory cortices, where it is hypothesized to structure synaptic connections between neurons. Previous studies have demonstrated how STDP can capture spiking information at short timescales using specific input configurations, such as coincident spiking, spike patterns and oscillatory spike trains. However, the corresponding computation in the case of arbitrary input signals is still unclear. This paper provides an overarching picture of the algorithm inherent to STDP, tying together many previous results for commonly used models of pairwise STDP. For a single neuron with plastic excitatory synapses, we show how STDP performs a spectral analysis on the temporal cross-correlograms between its afferent spike trains. The postsynaptic responses and STDP learning window determine kernel functions that specify how the neuron "sees" the input correlations. We thus denote this unsupervised learning scheme as 'kernel spectral component analysis' (kSCA). In particular, the whole input correlation structure must be considered since all plastic synapses compete with each other. We find that kSCA is enhanced when weight-dependent STDP induces gradual synaptic competition. For a spiking neuron with a "linear" response and pairwise STDP alone, we find that kSCA resembles principal component analysis (PCA). However, plain STDP does not isolate correlation sources in general, e.g., when they are mixed among the input spike trains. In other words, it does not perform independent component analysis (ICA). Tuning the neuron to a single correlation source can be achieved when STDP is paired with a homeostatic mechanism that reinforces the competition between synaptic inputs. Our results suggest that neuronal networks equipped with STDP can process signals encoded in the transient spiking activity at the timescales of tens of milliseconds for usual STDP.     

Consequences of Converting Graded to Action Potentials upon Neural Information Coding and Energy Efficiency

Information is encoded in neural circuits using both graded and action potentials, converting between them within single neurons and successive processing layers. This conversion is accompanied by information loss and a drop in energy efficiency. We investigate the biophysical causes of this loss of information and efficiency by comparing spiking neuron models, containing stochastic voltage-gated Na⁺ and K⁺ channels, with generator potential and graded potential models lacking voltage-gated Na⁺ channels. We identify three causes of information loss in the generator potential that are the by-product of action potential generation: (1) the voltage-gated Na⁺ channels necessary for action potential generation increase intrinsic noise and (2) introduce non-linearities, and (3) the finite duration of the action potential creates a ‘footprint’ in the generator potential that obscures incoming signals. These three processes reduce information rates by ∼50% in generator potentials, to ∼3 times that of spike trains. Both generator potentials and graded potentials consume almost an order of magnitude less energy per second than spike trains. Because of the lower information rates of generator potentials they are substantially less energy efficient than graded potentials. However, both are an order of magnitude more efficient than spike trains due to the higher energy costs and low information content of spikes, emphasizing that there is a two-fold cost of converting analogue to digital; information loss and cost inflation.     

Bayesian nonparametric analysis for the detection of spikes in noisy calcium imaging data

Recent advancements in miniaturized fluorescence microscopy have made it possible to investigate neuronal responses to external stimuli in awake behaving animals through the analysis of intracellular calcium signals. An ongoing challenge is deconvolving the temporal signals to extract the spike trains from the noisy calcium signals' time series. In this article, we propose a nested Bayesian finite mixture specification that allows the estimation of spiking activity and, simultaneously, reconstructing the distributions of the calcium transient spikes' amplitudes under different experimental conditions. The proposed model leverages two nested layers of random discrete mixture priors to borrow information between experiments and discover similarities in the distributional patterns of neuronal responses to different stimuli. Furthermore, the spikes' intensity values are also clustered within and between experimental conditions to determine the existence of common (recurring) response amplitudes. Simulation studies and the analysis of a dataset from the Allen Brain Observatory show the effectiveness of the method in clustering and detecting neuronal activities.     

Imaging the environment of green fluorescent protein

An emerging theme in cell biology is that cell surface receptors need to be considered as part of supramolecular complexes of proteins and lipids facilitating specific receptor conformations and distinct distributions, e.g., at the immunological synapse. Thus, a new goal is to develop bioimaging that not only locates proteins in live cells but can also probe their environment. Such a technique is demonstrated here using fluorescence lifetime imaging of green fluorescent protein (GFP). We first show, by time-correlated single-photon counting, that the fluorescence decay of GFP depends on the local refractive index. This is in agreement with the Strickler Berg formula, relating the Einstein A and B coefficients for absorption and spontaneous emission in molecules. We then quantitatively image, by wide-field time-gated fluorescence lifetime imaging, the refractive index of the environment of GFP. This novel approach paves the way for imaging the biophysical environment of specific GFP-tagged proteins in live cells.     

Firing-rate models capture essential response dynamics of LGN relay cells

Firing-rate models provide a practical tool for studying signal processing in the early visual system, permitting more thorough mathematical analysis than spike-based models. We show here that essential response properties of relay cells in the lateral geniculate nucleus (LGN) can be captured by surprisingly simple firing-rate models consisting of a low-pass filter and a nonlinear activation function. The starting point for our analysis are two spiking neuron models based on experimental data: a spike-response model fitted to data from macaque (Carandini et al. J. Vis., 20(14), 1–2011, 2007), and a model with conductance-based synapses and afterhyperpolarizing currents fitted to data from cat (Casti et al. J. Comput. Neurosci., 24(2), 235–252, 2008). We obtained the nonlinear activation function by stimulating the model neurons with stationary stochastic spike trains, while we characterized the linear filter by fitting a low-pass filter to responses to sinusoidally modulated stochastic spike trains. To account for the non-Poisson nature of retinal spike trains, we performed all analyses with spike trains with higher-order gamma statistics in addition to Poissonian spike trains. Interestingly, the properties of the low-pass filter depend only on the average input rate, but not on the modulation depth of sinusoidally modulated input. Thus, the response properties of our model are fully specified by just three parameters (low-frequency gain, cutoff frequency, and delay) for a given mean input rate and input regularity. This simple firing-rate model reproduces the response of spiking neurons to a step in input rate very well for Poissonian as well as for non-Poissonian input. We also found that the cutoff frequencies, and thus the filter time constants, of the rate-based model are unrelated to the membrane time constants of the underlying spiking models, in agreement with similar observations for simpler models.     

Response dynamics and receptive-field organization of catfish ganglion cells [published erratum appears in J Gen Physiol 1995 Aug;106(2):following 388]

Responses from catfish retinal ganglion cells were evoked by a spot or an annulus of light and were analyzed by a procedure identical to the one used previously to study catfish amacrine cells (Sakai H. M., and K.-I. Naka, 1992. Journal of Neurophysiology. 67:430-442.). In two- input white-noise experiments, a response evoked by simultaneous stimulation of the center and surround was decomposed into the components generated by the center and surround through a process of cross-correlation. The center and surround responses were also decomposed into their linear and nonlinear components so that the response dynamics of the linear and nonlinear components could be measured. We found that the concentric organization of the receptive field was determined by linear components, i.e., the first-order kernels generated by the center and surround were of opposite polarity. Both the center and surround generated second-order kernels with similar signatures, i.e., the second-order components formed a monotonic receptive field. The peak response time of the first- and second-order kernels from the surround was longer by approximately 20 ms than that of the center. Except for the DC potential present in the intracellular responses, almost identical first- and second-order kernels for the center and surround were obtained from both the intracellular response and spike discharges. Thus, information on concentric organization of a receptive field is translated into spike discharges with little loss of information. A train of spike discharges carries, simultaneously, at least four kinds of information: two linear and two nonlinear components, which originate in the receptive field center and the surround. A spike train is not a simple signaling device but is a carrier of complex and multiple signals. Victor, J. D., and R. M. Shapley (1979. Journal of General Physiology. 74:671-687.) discovered similarly that, in the cat retina, static second-order nonlinearity is encoded into spike trains. Results obtained in this study support the thesis that signals generated by the preganglionic cells are translated into spike discharges without major modification and that those signals can be recovered from the spike trains (Sakuranaga, M., Y. Ando, and K.-I. Naka. 1987. Journal of General Physiology. 90:229-259.; Korenberg, M. J., H. M. Sakai, and K.-I. Naka. 1989. Journal of Neurophysiology. 61:1110-1120.). Current injection studies have shown that such signal transmission is possible (Sakai, H. M., and K.-I. Naka, 1988a. Journal of Neurophysiology. 60:1549-1567.; 1990. Journal of Neurophysiology. 63:105-119.).     

Pergressive Fourier analysis of encephalograms and renal potassium excretion data

Abstract

Through pergressive Fourier analysis of samples typical properties of this method are demonstrated. The most important difference from other mostly applied methods is that not only frequences (or period lengths, respectively) and amplitude values are used, but also phase relations of hidden oscillations, analogously to optical imaging when, e.g., a laser is added to lenses so that also phase relations of the wave trains in the optical field are used. In this way, from a three dimensional object not a plane picture is produced, but a picture equivalent to the original.

In the pergressive Fourier analysis initial phase and period length of hidden oscillations are accurately obtained through linear regression in a phase diagram, regardless whether the at random selected analysing interval is an integer multiple of the hidden period length, or not. As compared to spectrum and correlation methods, pergressive Fourier analysis produces a consideraly higher accuracy in the calculation of oscillation parameters.     

Event Ordering in Live-Cell Imaging Determined from Temporal Cross-Correlation Asymmetry

We use the temporal asymmetry of the cross-correlation function to determine the temporal ordering of spatially localized cellular events in live-cell multichannel fluorescence imaging. The analysis is well suited to noisy, stochastic systems where the temporal order may not be apparent in the raw data. The approach is applicable to any biochemical reaction not in chemical equilibrium, including protein complex assembly, sequential enzymatic processes, gene regulation, and other cellular signaling events. As an automated quantitative measure, this approach allows the data to be readily interpreted statistically with minimal subjective biases. We first test the technique using simulations of simple biophysical models with a definite temporal ordering. We then demonstrate the approach by extracting the temporal ordering of three proteins—actin, sorting nexin 9, and clathrin—in the endocytic pathway.