A dynamical model of C. elegans thermal preference reveals independent excitatory and inhibitory learning pathways.

Caenorhabditis elegans is capable of learning and remembering behaviorally relevant cues such as smells, tastes, and temperature. This is an example of associative learning, a process in which behavior is modified by making associations between various stimuli. Since the mathematical theory of conditioning does not account for some of its salient aspects, such as spontaneous recovery of extinguished associations, accurate modeling of behavior of real animals during conditioning has turned out difficult. Here, we do this in the context of the dynamics of the thermal preference of C. elegans. We quantify C. elegans thermotaxis in response to various conditioning temperatures, starvation durations, and genetic perturbations using a high-resolution microfluidic droplet assay. We model these data comprehensively, within a biologically interpretable, multi-modal framework. We find that the strength of the thermal preference is composed of two independent, genetically separable contributions and requires a model with at least four dynamical variables. One pathway positively associates the experienced temperature independently of food and the other negatively associates with the temperature when food is absent. The multidimensional structure of the association strength provides an explanation for the apparent classical temperature-food association of C. elegans thermal preference and a number of longstanding questions in animal learning, including spontaneous recovery, asymmetric response to appetitive vs. aversive cues, latent inhibition, and generalization among similar cues.

Data Efficiency, Dimensionality Reduction, and the Generalized Symmetric Information Bottleneck.

The symmetric information bottleneck (SIB), an extension of the more familiar information bottleneck, is a dimensionality-reduction technique that simultaneously compresses two random variables to preserve information between their compressed versions. We introduce the generalized symmetric information bottleneck (GSIB), which explores different functional forms of the cost of such simultaneous reduction. We then explore the data set size requirements of such simultaneous compression. We do this by deriving bounds and root-mean-squared estimates of statistical fluctuations of the involved loss functions. We show that in typical situations, the simultaneous GSIB compression requires qualitatively less data to achieve the same errors compared to compressing variables one at a time. We suggest that this is an example of a more general principle that simultaneous compression is more data efficient than independent compression of each of the input variables.

Constrained brain volume in an efficient coding model explains the fraction of excitatory and inhibitory neurons in sensory cortices.

The number of neurons in mammalian cortex varies by multiple orders of magnitude across different species. In contrast, the ratio of excitatory to inhibitory neurons (E:I ratio) varies in a much smaller range, from 3:1 to 9:1 and remains roughly constant for different sensory areas within a species. Despite this structure being important for understanding the function of neural circuits, the reason for this consistency is not yet understood. While recent models of vision based on the efficient coding hypothesis show that increasing the number of both excitatory and inhibitory cells improves stimulus representation, the two cannot increase simultaneously due to constraints on brain volume. In this work, we implement an efficient coding model of vision under a constraint on the volume (using number of neurons as a surrogate) while varying the E:I ratio. We show that the performance of the model is optimal at biologically observed E:I ratios under several metrics. We argue that this happens due to trade-offs between the computational accuracy and the representation capacity for natural stimuli. Further, we make experimentally testable predictions that 1) the optimal E:I ratio should be higher for species with a higher sparsity in the neural activity and 2) the character of inhibitory synaptic distributions and firing rates should change depending on E:I ratio. Our findings, which are supported by our new preliminary analyses of publicly available data, provide the first quantitative and testable hypothesis based on optimal coding models for the distribution of excitatory and inhibitory neural types in the mammalian sensory cortices.

Inferring phenomenological models of first passage processes.

Biochemical processes in cells are governed by complex networks of many chemical species interacting stochastically in diverse ways and on different time scales. Constructing microscopically accurate models of such networks is often infeasible. Instead, here we propose a systematic framework for building phenomenological models of such networks from experimental data, focusing on accurately approximating the time it takes to complete the process, the First Passage (FP) time. Our phenomenological models are mixtures of Gamma distributions, which have a natural biophysical interpretation. The complexity of the models is adapted automatically to account for the amount of available data and its temporal resolution. The framework can be used for predicting behavior of FP systems under varying external conditions. To demonstrate the utility of the approach, we build models for the distribution of inter-spike intervals of a morphologically complex neuron, a Purkinje cell, from experimental and simulated data. We demonstrate that the developed models can not only fit the data, but also make nontrivial predictions. We demonstrate that our coarse-grained models provide constraints on more mechanistically accurate models of the involved phenomena.

Automated, predictive, and interpretable inference of Caenorhabditis elegans escape dynamics.

The roundworm Caenorhabditis elegans exhibits robust escape behavior in response to rapidly rising temperature. The behavior lasts for a few seconds, shows history dependence, involves both sensory and motor systems, and is too complicated to model mechanistically using currently available knowledge. Instead we model the process phenomenologically, and we use the Sir Isaac dynamical inference platform to infer the model in a fully automated fashion directly from experimental data. The inferred model requires incorporation of an unobserved dynamical variable and is biologically interpretable. The model makes accurate predictions about the dynamics of the worm behavior, and it can be used to characterize the functional logic of the dynamical system underlying the escape response. This work illustrates the power of modern artificial intelligence to aid in discovery of accurate and interpretable models of complex natural systems.

Latent Dynamical Variables Produce Signatures of Spatiotemporal Criticality in Large Biological Systems.

Understanding the activity of large populations of neurons is difficult due to the combinatorial complexity of possible cell-cell interactions. To reduce the complexity, coarse graining had been previously applied to experimental neural recordings, which showed over two decades of apparent scaling in free energy, activity variance, eigenvalue spectra, and correlation time, hinting that the mouse hippocampus operates in a critical regime. We model such data by simulating conditionally independent binary neurons coupled to a small number of long-timescale stochastic fields and then replicating the coarse-graining procedure and analysis. This reproduces the experimentally observed scalings, suggesting that they do not require fine-tuning of internal parameters, but will arise in any system, biological or not, where activity variables are coupled to latent dynamic stimuli. Parameter sweeps for our model suggest that emergence of scaling requires most of the cells in a population to couple to the latent stimuli, predicting that even the celebrated place cells must also respond to nonplace stimuli.

Randomly connected networks generate emergent selectivity and predict decoding properties of large populations of neurons.

Modern recording methods enable sampling of thousands of neurons during the performance of behavioral tasks, raising the question of how recorded activity relates to theoretical models. In the context of decision making, functional connectivity between choice-selective cortical neurons was recently reported. The straightforward interpretation of these data suggests the existence of selective pools of inhibitory and excitatory neurons. Computationally investigating an alternative mechanism for these experimental observations, we find that a randomly connected network of excitatory and inhibitory neurons generates single-cell selectivity, patterns of pairwise correlations, and the same ability of excitatory and inhibitory populations to predict choice, as in experimental observations. Further, we predict that, for this task, there are no anatomically defined subpopulations of neurons representing choice, and that choice preference of a particular neuron changes with the details of the task. We suggest that distributed stimulus selectivity and functional organization in population codes could be emergent properties of randomly connected networks.

Chance, long tails, and inference in a non-Gaussian, Bayesian theory of vocal learning in songbirds.

Traditional theories of sensorimotor learning posit that animals use sensory error signals to find the optimal motor command in the face of Gaussian sensory and motor noise. However, most such theories cannot explain common behavioral observations, for example, that smaller sensory errors are more readily corrected than larger errors and large abrupt (but not gradually introduced) errors lead to weak learning. Here, we propose a theory of sensorimotor learning that explains these observations. The theory posits that the animal controls an entire probability distribution of motor commands rather than trying to produce a single optimal command and that learning arises via Bayesian inference when new sensory information becomes available. We test this theory using data from a songbird, the Bengalese finch, that is adapting the pitch (fundamental frequency) of its song following perturbations of auditory feedback using miniature headphones. We observe the distribution of the sung pitches to have long, non-Gaussian tails, which, within our theory, explains the observed dynamics of learning. Further, the theory makes surprising predictions about the dynamics of the shape of the pitch distribution, which we confirm experimentally.

Universal Properties of Concentration Sensing in Large Ligand-Receptor Networks.

Cells estimate concentrations of chemical ligands in their environment using a limited set of receptors. Recent work has shown that the temporal sequence of binding and unbinding events on just a single receptor can be used to estimate the concentrations of multiple ligands. Here, for a network of many ligands and many receptors, we show that such temporal sequences can be used to estimate the concentration of a few times as many ligand species as there are receptors. Crucially, we show that the spectrum of the inverse covariance matrix of these estimates has several universal properties, which we trace to properties of Vandermonde matrices. We argue that this can be used by cells in realistic biochemical decoding networks.

Motor control by precisely timed spike patterns.

A fundamental problem in neuroscience is understanding how sequences of action potentials ("spikes") encode information about sensory signals and motor outputs. Although traditional theories assume that this information is conveyed by the total number of spikes fired within a specified time interval (spike rate), recent studies have shown that additional information is carried by the millisecond-scale timing patterns of action potentials (spike timing). However, it is unknown whether or how subtle differences in spike timing drive differences in perception or behavior, leaving it unclear whether the information in spike timing actually plays a role in brain function. By examining the activity of individual motor units (the muscle fibers innervated by a single motor neuron) and manipulating patterns of activation of these neurons, we provide both correlative and causal evidence that the nervous system uses millisecond-scale variations in the timing of spikes within multispike patterns to control a vertebrate behavior-namely, respiration in the Bengalese finch, a songbird. These findings suggest that a fundamental assumption of current theories of motor coding requires revision.

Physical Limit to Concentration Sensing in a Changing Environment.

Cells adapt to changing environments by sensing ligand concentrations using specific receptors. The accuracy of sensing is ultimately limited by the finite number of ligand molecules bound by receptors. Previously derived physical limits to sensing accuracy largely have assumed that the concentration was constant and ignored its temporal fluctuations. We formulate the problem of concentration sensing in a strongly fluctuating environment as a nonlinear field-theoretic problem, for which we find an excellent approximate Gaussian solution. We derive a new physical bound on the relative error in concentration c which scales as δc/c∼(Dacτ)^{-1/4} with ligand diffusivity D, receptor cross section a, and characteristic fluctuation timescale τ, in stark contrast to the usual Berg and Purcell bound δc/c∼(DacT)^{-1/2} for a perfect receptor sensing concentration during time T. We show how the bound can be achieved by a biochemical network downstream of the receptor that adapts the kinetics of signaling as a function of the square root of the sensed concentration.

Limits to the precision of gradient sensing with spatial communication and temporal integration.

Gradient sensing requires at least two measurements at different points in space. These measurements must then be communicated to a common location to be compared, which is unavoidably noisy. Although much is known about the limits of measurement precision by cells, the limits placed by the communication are not understood. Motivated by recent experiments, we derive the fundamental limits to the precision of gradient sensing in a multicellular system, accounting for communication and temporal integration. The gradient is estimated by comparing a "local" and a "global" molecular reporter of the external concentration, where the global reporter is exchanged between neighboring cells. Using the fluctuation-dissipation framework, we find, in contrast to the case when communication is ignored, that precision saturates with the number of cells independently of the measurement time duration, because communication establishes a maximum length scale over which sensory information can be reliably conveyed. Surprisingly, we also find that precision is improved if the local reporter is exchanged between cells as well, albeit more slowly than the global reporter. The reason is that whereas exchange of the local reporter weakens the comparison, it decreases the measurement noise. We term such a model "regional excitation-global inhibition." Our results demonstrate that fundamental sensing limits are necessarily sharpened when the need to communicate information is taken into account.

Cell-cell communication enhances the capacity of cell ensembles to sense shallow gradients during morphogenesis.

Collective cell responses to exogenous cues depend on cell-cell interactions. In principle, these can result in enhanced sensitivity to weak and noisy stimuli. However, this has not yet been shown experimentally, and little is known about how multicellular signal processing modulates single-cell sensitivity to extracellular signaling inputs, including those guiding complex changes in the tissue form and function. Here we explored whether cell-cell communication can enhance the ability of cell ensembles to sense and respond to weak gradients of chemotactic cues. Using a combination of experiments with mammary epithelial cells and mathematical modeling, we find that multicellular sensing enables detection of and response to shallow epidermal growth factor (EGF) gradients that are undetectable by single cells. However, the advantage of this type of gradient sensing is limited by the noisiness of the signaling relay, necessary to integrate spatially distributed ligand concentration information. We calculate the fundamental sensory limits imposed by this communication noise and combine them with the experimental data to estimate the effective size of multicellular sensory groups involved in gradient sensing. Functional experiments strongly implicated intercellular communication through gap junctions and calcium release from intracellular stores as mediators of collective gradient sensing. The resulting integrative analysis provides a framework for understanding the advantages and limitations of sensory information processing by relays of chemically coupled cells.

Simple biochemical networks allow accurate sensing of multiple ligands with a single receptor.

Cells use surface receptors to estimate concentrations of external ligands. Limits on the accuracy of such estimations have been well studied for pairs of ligand and receptor species. However, the environment typically contains many ligands, which can bind to the same receptors with different affinities, resulting in cross-talk. In traditional rate models, such cross-talk prevents accurate inference of concentrations of individual ligands. In contrast, here we show that knowing the precise timing sequence of stochastic binding and unbinding events allows one receptor to provide information about multiple ligands simultaneously and with a high accuracy. We show that such high-accuracy estimation of multiple concentrations can be realized with simple structural modifications of the familiar kinetic proofreading biochemical network diagram. We give two specific examples of such modifications. We argue that structural and functional features of real cellular biochemical sensory networks in immune cells, such as feedforward and feedback loops or ligand antagonism, sometimes can be understood as solutions to the accurate multi-ligand estimation problem.

Growth of bacteria in 3-d colonies.

The dynamics of growth of bacterial populations has been extensively studied for planktonic cells in well-agitated liquid culture, in which all cells have equal access to nutrients. In the real world, bacteria are more likely to live in physically structured habitats as colonies, within which individual cells vary in their access to nutrients. The dynamics of bacterial growth in such conditions is poorly understood, and, unlike that for liquid culture, there is not a standard broadly used mathematical model for bacterial populations growing in colonies in three dimensions (3-d). By extending the classic Monod model of resource-limited population growth to allow for spatial heterogeneity in the bacterial access to nutrients, we develop a 3-d model of colonies, in which bacteria consume diffusing nutrients in their vicinity. By following the changes in density of E. coli in liquid and embedded in glucose-limited soft agar, we evaluate the fit of this model to experimental data. The model accounts for the experimentally observed presence of a sub-exponential, diffusion-limited growth regime in colonies, which is absent in liquid cultures. The model predicts and our experiments confirm that, as a consequence of inter-colony competition for the diffusing nutrients and of cell death, there is a non-monotonic relationship between total number of colonies within the habitat and the total number of individual cells in all of these colonies. This combined theoretical-experimental study reveals that, within 3-d colonies, E. coli cells are loosely packed, and colonies produce about 2.5 times as many cells as the liquid culture from the same amount of nutrients. We verify that this is because cells in liquid culture are larger than in colonies. Our model provides a baseline description of bacterial growth in 3-d, deviations from which can be used to identify phenotypic heterogeneities and inter-cellular interactions that further contribute to the structure of bacterial communities.

Single variant bottleneck in the early dynamics of H. influenzae bacteremia in neonatal rats questions the theory of independent action.

There is an abundance of information about the genetic basis, physiological and molecular mechanisms of bacterial pathogenesis. In contrast, relatively little is known about population dynamic processes, by which bacteria colonize hosts and invade tissues and cells and thereby cause disease. In an article published in 1978, Moxon and Murphy presented evidence that, when inoculated intranasally with a mixture streptomycin sensitive and resistant (Sm S and Sm R ) and otherwise isogenic strains of Haemophilus influenzae type b (Hib), neonatal rats develop a bacteremic infection that often is dominated by only one strain, Sm S or Sm R . After ruling out other possibilities through years of related experiments, the field seems to have settled on a plausible explanation for this phenomenon: the first bacterium to invade the host activates the host immune response that 'shuts the door' on the second invading strain. To explore this hypothesis in a necessarily quantitative way, we modeled this process with a set of mixed stochastic and deterministic differential equations. Our analysis of the properties of this model with realistic parameters suggests that this hypothesis cannot explain the experimental results of Moxon and Murphy, and in particular the observed relationship between the frequency of different types of blood infections (bacteremias) and the inoculum size. We propose modifications to the model that come closer to explaining these data. However, the modified and better fitting model contradicts the common theory of independent action of individual bacteria in establishing infections. We suggest possible experiments that would be able to confirm or reject our proposed modification of the early infection model.

Luria-Delbrück, revisited: the classic experiment does not rule out Lamarckian evolution.

We re-examined data from the classic Luria-Delbrück fluctuation experiment, which is often credited with establishing a Darwinian basis for evolution. We argue that, for the Lamarckian model of evolution to be ruled out by the experiment, the experiment must favor pure Darwinian evolution over both the Lamarckian model and a model that allows both Darwinian and Lamarckian mechanisms (as would happen for bacteria with CRISPR-Cas immunity). Analysis of the combined model was not performed in the original 1943 paper. The Luria-Delbrück paper also did not consider the possibility of neither model fitting the experiment. Using Bayesian model selection, we find that the Luria-Delbrück experiment, indeed, favors the Darwinian evolution over purely Lamarckian. However, our analysis does not rule out the combined model, and hence cannot rule out Lamarckian contributions to the evolutionary dynamics.

Millisecond-scale motor encoding in a cortical vocal area.

Studies of motor control have almost universally examined firing rates to investigate how the brain shapes behavior. In principle, however, neurons could encode information through the precise temporal patterning of their spike trains as well as (or instead of) through their firing rates. Although the importance of spike timing has been demonstrated in sensory systems, it is largely unknown whether timing differences in motor areas could affect behavior. We tested the hypothesis that significant information about trial-by-trial variations in behavior is represented by spike timing in the songbird vocal motor system. We found that neurons in motor cortex convey information via spike timing far more often than via spike rate and that the amount of information conveyed at the millisecond timescale greatly exceeds the information available from spike counts. These results demonstrate that information can be represented by spike timing in motor circuits and suggest that timing variations evoke differences in behavior.

Stereotypical Escape Behavior in Caenorhabditis elegans Allows Quantification of Effective Heat Stimulus Level.

A goal of many sensorimotor studies is to quantify the stimulus-behavioral response relation for specific organisms and specific sensory stimuli. This is especially important to do in the context of painful stimuli since most animals in these studies cannot easily communicate to us their perceived levels of such noxious stimuli. Thus progress on studies of nociception and pain-like responses in animal models depends crucially on our ability to quantitatively and objectively infer the sensed levels of these stimuli from animal behaviors. Here we develop a quantitative model to infer the perceived level of heat stimulus from the stereotyped escape response of individual nematodes Caenorhabditis elegans stimulated by an IR laser. The model provides a method for quantification of analgesic-like effects of chemical stimuli or genetic mutations in C. elegans. We test ibuprofen-treated worms and a TRPV (transient receptor potential) mutant, and we show that the perception of heat stimuli for the ibuprofen treated worms is lower than the wild-type. At the same time, our model shows that the mutant changes the worm's behavior beyond affecting the thermal sensory system. Finally, we determine the stimulus level that best distinguishes the analgesic-like effects and the minimum number of worms that allow for a statistically significant identification of these effects.

Role of spatial averaging in multicellular gradient sensing.

Gradient sensing underlies important biological processes including morphogenesis, polarization, and cell migration. The precision of gradient sensing increases with the length of a detector (a cell or group of cells) in the gradient direction, since a longer detector spans a larger range of concentration values. Intuition from studies of concentration sensing suggests that precision should also increase with detector length in the direction transverse to the gradient, since then spatial averaging should reduce the noise. However, here we show that, unlike for concentration sensing, the precision of gradient sensing decreases with transverse length for the simplest gradient sensing model, local excitation-global inhibition. The reason is that gradient sensing ultimately relies on a subtraction of measured concentration values. While spatial averaging indeed reduces the noise in these measurements, which increases precision, it also reduces the covariance between the measurements, which results in the net decrease in precision. We demonstrate how a recently introduced gradient sensing mechanism, regional excitation-global inhibition (REGI), overcomes this effect and recovers the benefit of transverse averaging. Using a REGI-based model, we compute the optimal two- and three-dimensional detector shapes, and argue that they are consistent with the shapes of naturally occurring gradient-sensing cell populations.