Testing the state-dependent model of time perception against experimental evidence

Coordinated movements, speech and other actions are impossible without precise timing. Realistic computational models of interval timing in the mammalian brain are expected to provide key insights into the underlying mechanisms of timing. Existing computational models of time perception have only been partially replicating experimental observations, such as the linear increase of time, the dopaminergic modulation of this increase, and the scalar property, i.e., the linear increase of the standard deviation of temporal estimates. In this work, we incorporate the state-dependent computational model, which encodes time in the dynamic evolution of network states without the need for a specific network structure into a biologically plausible prefrontal cortex (PFC) model based on in vivo and in vitro recordings of rodents. Specifically, we stimulated 1000 neurons in the beginning and in the end of a range of different time intervals, extracted states of neurons and trained the readout layer based on these states using least squares to predict the respective inter stimulus interval. We show that the naturally occurring heterogeneity in cellular and synaptic parameters in the PFC is sufficient to encode time over several hundreds of milliseconds. The readout faithfully represents the duration between two stimuli applied to the superficial layers of the network, thus fulfilling the requirement of a linear encoding of time. A simulated activation of the D2 dopamine receptor leads to an overestimation and an inactivation to an underestimation of time, in line with experimental results. Furthermore, we show that the scalar property holds true for intervals of several hundred milliseconds, and provide a mechanistic explanation for the origin of the scalar property as well as its deviations. We conclude that this model can represent durations up to 750 ms in a biophysically plausible setting, compatible with experimental findings in this regime.

On the sparsity of fitness functions and implications for learning

Fitness functions map biological sequences to a scalar property of interest. Accurate estimation of these functions yields biological insight and sets the foundation for model-based sequence design. However, the fitness datasets available to learn these functions are typically small relative to the large combinatorial space of sequences; characterizing how much data are needed for accurate estimation remains an open problem. There is a growing body of evidence demonstrating that empirical fitness functions display substantial sparsity when represented in terms of epistatic interactions. Moreover, the theory of Compressed Sensing provides scaling laws for the number of samples required to exactly recover a sparse function. Motivated by these results, we develop a framework to study the sparsity of fitness functions sampled from a generalization of the NK model, a widely used random field model of fitness functions. In particular, we present results that allow us to test the effect of the Generalized NK (GNK) model’s interpretable parameters—sequence length, alphabet size, and assumed interactions between sequence positions—on the sparsity of fitness functions sampled from the model and, consequently, the number of measurements required to exactly recover these functions. We validate our framework by demonstrating that GNK models with parameters set according to structural considerations can be used to accurately approximate the number of samples required to recover two empirical protein fitness functions and an RNA fitness function. In addition, we show that these GNK models identify important higher-order epistatic interactions in the empirical fitness functions using only structural information.

Signal neutrality, scalar property, and collapsing boundaries as consequences of a learned multi-time scale strategy

Experiments and models in perceptual decision-making point to a key role of an integration process that accumulates sensory evidence over time. We endow a probabilistic agent comprising several such integrators with widely spread time scales and let it learn, by trial-and-error, to weight the different filtered versions of a noisy signal. The agent discovers a strategy markedly different from the literature "standard", according to which a decision made when the accumulated evidence hits a predetermined threshold. The agent instead decides during fleeting windows corresponding to the alignment of many integrators, akin to a majority vote. This strategy presents three distinguishing signatures. 1) Signal neutrality: a marked insensitivity to the signal coherence in the interval preceding the decision, as also observed in experiments. 2) Scalar property: the mean of the response times varies glaringly for different signal coherences, yet the shape of the distribution stays largely unchanged. 3) Collapsing boundaries: the agent learns to behave as if subject to a non-monotonic urgency signal, reminiscent in shape of the theoretically optimal. These three characteristics, which emerge from the interaction of a multi-scale learning agent with a highly volatile environment, are hallmarks, we argue, of an optimal decision strategy in challenging situations. As such, the present results may shed light on general information-processing principles leveraged by the brain itself.

On the sparsity of fitness functions and implications for learning

Fitness functions map biological sequences to a scalar property of interest. Accurate estimation of these functions yields biological insight and sets the foundation for model-based sequence design. However, the amount of fitness data available to learn these functions is typically small relative to the large combinatorial space of sequences; characterizing how much data is needed for accurate estimation remains an open problem. There is a growing body of evidence demonstrating that empirical fitness functions display substantial sparsity when represented in terms of epistatic interactions. Moreover, the theory of Compressed Sensing provides scaling laws for the number of samples required to exactly recover a sparse function. Motivated by these results, we study the sparsity of fitness functions sampled from a generalization of the NK model, a widely-used random field model of fitness functions. In particular, we present theoretical results that allow us to test the effect of the Generalized NK (GNK) model's interpretable parameters---sequence length, alphabet size, and assumed interactions between sequence positions---on the sparsity of fitness functions sampled from the model and, consequently, the number of measurements required to exactly recover these functions. Further, we show that GNK fitness functions with parameters set according to protein structural contacts can be used to accurately approximate the number of samples required to estimate two empirical protein fitness functions, and are able to identify important higher-order epistatic interactions in these functions using only structural information.

Physical Laws Shape Up HOX Gene Collinearity

Hox gene collinearity (HGC) is a multi-scalar property of many animal phyla particularly important in embryogenesis. It relates entities and events occurring in Hox clusters inside the chromosome DNA and in embryonic tissues. These two entities differ in linear size by more than four orders of magnitude. HGC is observed as spatial collinearity (SC), where the Hox genes are located in the order (Hox1, Hox2, Hox3 …) along the 3′ to 5′ direction of DNA in the genome and a corresponding sequence of ontogenetic units (E1, E2, E3, …) located along the Anterior—Posterior axis of the embryo. Expression of Hox1 occurs in E1, Hox2 in E2, Hox3 in E3, etc. Besides SC, a temporal collinearity (TC) has been also observed in many vertebrates. According to TC, first Hox1 is expressed in E1; later, Hox2 is expressed in E2, followed by Hox3 in E3, etc. Lately, doubt has been raised about whether TC really exists. A biophysical model (BM) was formulated and tested during the last 20 years. According to BM, physical forces are created which pull the Hox genes one after the other, driving them to a transcription factory domain where they are transcribed. The existing experimental data support this BM description. Symmetry is a physical–mathematical property of matter that was explored in depth by Noether who formulated a ground-breaking theory (NT) that applies to all sizes of matter. NT may be applied to biology in order to explain the origin of HGC in animals developing not only along the A/P axis, but also to animals with circular symmetry.

Respiration and brain neural dynamics associated with interval timing during odor fear learning in rats

In fear conditioning, where a conditioned stimulus predicts the arrival of an aversive stimulus, the animal encodes the time interval between the two stimuli. Here we monitored respiration to visualize anticipatory behavioral responses in an odor fear conditioning in rats, while recording theta (5–15 Hz) and gamma (40–80 Hz) brain oscillatory activities in the medial prefrontal cortex (mPFC), basolateral amygdala (BLA), dorsomedial striatum (DMS) and olfactory piriform cortex (PIR). We investigated the temporal patterns of respiration frequency and of theta and gamma activity power during the odor-shock interval, comparing two interval durations. We found that akin to respiration patterns, theta temporal curves were modulated by the duration of the odor-shock interval in the four recording sites, and respected scalar property in mPFC and DMS. In contrast, gamma temporal curves were modulated by the interval duration only in the mPFC, and in a manner that did not respect scalar property. This suggests a preferential role for theta rhythm in interval timing. In addition, our data bring the novel idea that the respiratory rhythm might take part in the setting of theta activity dynamics related to timing.

Respiration and brain neural dynamics associated with interval timing during odor fear learning in rats

ABSTRACTIn fear conditioning, where a conditioned stimulus predicts the arrival of an aversive stimulus, the animal encodes the time interval between the two stimuli. Freezing, the most used index to assess learned fear, lacks the temporal resolution required to investigate interval timing at the early stages of learning. Here we monitored respiration to visualize anticipatory behavioral responses in an odor fear conditioning in rats, while recording theta (5-15Hz) and gamma (40-80Hz) brain oscillatory activities in the medial prefrontal cortex (mPFC), basolateral amygdala (BLA), dorsomedial striatum (DMS) and olfactory piriform cortex (PIR). We investigated the temporal patterns of respiration frequency and of theta and gamma activity power during the odor-shock interval. We found that akin to respiration patterns, theta temporal curves were modulated by the duration of the odor-shock interval in the four recording sites, and respected scalar property in mPFC and DMS. In contrast, gamma temporal curves were modulated by the interval duration only in the mPFC, and in a manner that did not respect scalar property. This suggests a preferential role for theta rhythm in interval timing. In addition, our data bring the novel idea that the respiratory rhythm might take part in the setting of theta activity dynamics.

Weber’s Law and the Scalar Property of Timing: A Test of Canine Timing

Domestic dogs completed a temporal bisection procedure that required a response to one lever following a light stimulus of short duration and to another lever following a light stimulus of a longer duration. The short and long durations across the four conditions were (0.5–2.0 s, 1.0–4.0 s, 2.0–8.0 s, and 4.0–16.0 s). Durations that were intermediate, the training durations, and the training durations, were presented during generalization tests. The dogs bisected the intervals near the geometric mean of the short and long-stimulus pair. Weber fractions were not constant when plotted as a function of time: A U-shaped function described them. These results replicate the findings of previous research reporting points of subjective equality falling close to the geometric mean and also confirm recent reports of systematic departures from Weber’s law.