TY - JOUR
AB - Motion tracking is a challenge the visual system has to solve by reading out the retinal population. It is still unclear how the information from different neurons can be combined together to estimate the position of an object. Here we recorded a large population of ganglion cells in a dense patch of salamander and guinea pig retinas while displaying a bar moving diffusively. We show that the bar’s position can be reconstructed from retinal activity with a precision in the hyperacuity regime using a linear decoder acting on 100+ cells. We then took advantage of this unprecedented precision to explore the spatial structure of the retina’s population code. The classical view would have suggested that the firing rates of the cells form a moving hill of activity tracking the bar’s position. Instead, we found that most ganglion cells in the salamander fired sparsely and idiosyncratically, so that their neural image did not track the bar. Furthermore, ganglion cell activity spanned an area much larger than predicted by their receptive fields, with cells coding for motion far in their surround. As a result, population redundancy was high, and we could find multiple, disjoint subsets of neurons that encoded the trajectory with high precision. This organization allows for diverse collections of ganglion cells to represent high-accuracy motion information in a form easily read out by downstream neural circuits.
AU - Marre, Olivier
AU - Botella Soler, Vicente
AU - Simmons, Kristina
AU - Mora, Thierry
AU - Tkacik, Gasper
AU - Berry, Michael
ID - 1697
IS - 7
JF - PLoS Computational Biology
TI - High accuracy decoding of dynamical motion from a large retinal population
VL - 11
ER -
TY - JOUR
AB - The activity of a neural network is defined by patterns of spiking and silence from the individual neurons. Because spikes are (relatively) sparse, patterns of activity with increasing numbers of spikes are less probable, but, with more spikes, the number of possible patterns increases. This tradeoff between probability and numerosity is mathematically equivalent to the relationship between entropy and energy in statistical physics. We construct this relationship for populations of up to N = 160 neurons in a small patch of the vertebrate retina, using a combination of direct and model-based analyses of experiments on the response of this network to naturalistic movies. We see signs of a thermodynamic limit, where the entropy per neuron approaches a smooth function of the energy per neuron as N increases. The form of this function corresponds to the distribution of activity being poised near an unusual kind of critical point. We suggest further tests of criticality, and give a brief discussion of its functional significance.
AU - Tkacik, Gasper
AU - Mora, Thierry
AU - Marre, Olivier
AU - Amodei, Dario
AU - Palmer, Stephanie
AU - Berry Ii, Michael
AU - Bialek, William
ID - 1701
IS - 37
JF - PNAS
TI - Thermodynamics and signatures of criticality in a network of neurons
VL - 112
ER -
TY - JOUR
AB - Bow-tie or hourglass structure is a common architectural feature found in many biological systems. A bow-tie in a multi-layered structure occurs when intermediate layers have much fewer components than the input and output layers. Examples include metabolism where a handful of building blocks mediate between multiple input nutrients and multiple output biomass components, and signaling networks where information from numerous receptor types passes through a small set of signaling pathways to regulate multiple output genes. Little is known, however, about how bow-tie architectures evolve. Here, we address the evolution of bow-tie architectures using simulations of multi-layered systems evolving to fulfill a given input-output goal. We find that bow-ties spontaneously evolve when the information in the evolutionary goal can be compressed. Mathematically speaking, bow-ties evolve when the rank of the input-output matrix describing the evolutionary goal is deficient. The maximal compression possible (the rank of the goal) determines the size of the narrowest part of the network—that is the bow-tie. A further requirement is that a process is active to reduce the number of links in the network, such as product-rule mutations, otherwise a non-bow-tie solution is found in the evolutionary simulations. This offers a mechanism to understand a common architectural principle of biological systems, and a way to quantitate the effective rank of the goals under which they evolved.
AU - Friedlander, Tamar
AU - Mayo, Avraham
AU - Tlusty, Tsvi
AU - Alon, Uri
ID - 1827
IS - 3
JF - PLoS Computational Biology
TI - Evolution of bow-tie architectures in biology
VL - 11
ER -
TY - JOUR
AB - Continuous-time Markov chains are commonly used in practice for modeling biochemical reaction networks in which the inherent randomness of themolecular interactions cannot be ignored. This has motivated recent research effort into methods for parameter inference and experiment design for such models. The major difficulty is that such methods usually require one to iteratively solve the chemical master equation that governs the time evolution of the probability distribution of the system. This, however, is rarely possible, and even approximation techniques remain limited to relatively small and simple systems. An alternative explored in this article is to base methods on only some low-order moments of the entire probability distribution. We summarize the theory behind such moment-based methods for parameter inference and experiment design and provide new case studies where we investigate their performance.
AU - Ruess, Jakob
AU - Lygeros, John
ID - 1861
IS - 2
JF - ACM Transactions on Modeling and Computer Simulation
TI - Moment-based methods for parameter inference and experiment design for stochastic biochemical reaction networks
VL - 25
ER -
TY - JOUR
AB - The concept of positional information is central to our understanding of how cells determine their location in a multicellular structure and thereby their developmental fates. Nevertheless, positional information has neither been defined mathematically nor quantified in a principled way. Here we provide an information-theoretic definition in the context of developmental gene expression patterns and examine the features of expression patterns that affect positional information quantitatively. We connect positional information with the concept of positional error and develop tools to directly measure information and error from experimental data. We illustrate our framework for the case of gap gene expression patterns in the early Drosophila embryo and show how information that is distributed among only four genes is sufficient to determine developmental fates with nearly single-cell resolution. Our approach can be generalized to a variety of different model systems; procedures and examples are discussed in detail.
AU - Tkacik, Gasper
AU - Dubuis, Julien
AU - Petkova, Mariela
AU - Gregor, Thomas
ID - 1885
IS - 1
JF - Genetics
TI - Positional information, positional error, and readout precision in morphogenesis: A mathematical framework
VL - 199
ER -
TY - JOUR
AB - We typically think of cells as responding to external signals independently by regulating their gene expression levels, yet they often locally exchange information and coordinate. Can such spatial coupling be of benefit for conveying signals subject to gene regulatory noise? Here we extend our information-theoretic framework for gene regulation to spatially extended systems. As an example, we consider a lattice of nuclei responding to a concentration field of a transcriptional regulator (the "input") by expressing a single diffusible target gene. When input concentrations are low, diffusive coupling markedly improves information transmission; optimal gene activation functions also systematically change. A qualitatively new regulatory strategy emerges where individual cells respond to the input in a nearly step-like fashion that is subsequently averaged out by strong diffusion. While motivated by early patterning events in the Drosophila embryo, our framework is generically applicable to spatially coupled stochastic gene expression models.
AU - Sokolowski, Thomas R
AU - Tkacik, Gasper
ID - 1940
IS - 6
JF - Physical Review E Statistical Nonlinear and Soft Matter Physics
TI - Optimizing information flow in small genetic networks. IV. Spatial coupling
VL - 91
ER -
TY - JOUR
AB - Systems biology rests on the idea that biological complexity can be better unraveled through the interplay of modeling and experimentation. However, the success of this approach depends critically on the informativeness of the chosen experiments, which is usually unknown a priori. Here, we propose a systematic scheme based on iterations of optimal experiment design, flow cytometry experiments, and Bayesian parameter inference to guide the discovery process in the case of stochastic biochemical reaction networks. To illustrate the benefit of our methodology, we apply it to the characterization of an engineered light-inducible gene expression circuit in yeast and compare the performance of the resulting model with models identified from nonoptimal experiments. In particular, we compare the parameter posterior distributions and the precision to which the outcome of future experiments can be predicted. Moreover, we illustrate how the identified stochastic model can be used to determine light induction patterns that make either the average amount of protein or the variability in a population of cells follow a desired profile. Our results show that optimal experiment design allows one to derive models that are accurate enough to precisely predict and regulate the protein expression in heterogeneous cell populations over extended periods of time.
AU - Ruess, Jakob
AU - Parise, Francesca
AU - Milias Argeitis, Andreas
AU - Khammash, Mustafa
AU - Lygeros, John
ID - 1538
IS - 26
JF - PNAS
TI - Iterative experiment design guides the characterization of a light-inducible gene expression circuit
VL - 112
ER -
TY - JOUR
AB - Many stochastic models of biochemical reaction networks contain some chemical species for which the number of molecules that are present in the system can only be finite (for instance due to conservation laws), but also other species that can be present in arbitrarily large amounts. The prime example of such networks are models of gene expression, which typically contain a small and finite number of possible states for the promoter but an infinite number of possible states for the amount of mRNA and protein. One of the main approaches to analyze such models is through the use of equations for the time evolution of moments of the chemical species. Recently, a new approach based on conditional moments of the species with infinite state space given all the different possible states of the finite species has been proposed. It was argued that this approach allows one to capture more details about the full underlying probability distribution with a smaller number of equations. Here, I show that the result that less moments provide more information can only stem from an unnecessarily complicated description of the system in the classical formulation. The foundation of this argument will be the derivation of moment equations that describe the complete probability distribution over the finite state space but only low-order moments over the infinite state space. I will show that the number of equations that is needed is always less than what was previously claimed and always less than the number of conditional moment equations up to the same order. To support these arguments, a symbolic algorithm is provided that can be used to derive minimal systems of unconditional moment equations for models with partially finite state space.
AU - Ruess, Jakob
ID - 1539
IS - 24
JF - Journal of Chemical Physics
TI - Minimal moment equations for stochastic models of biochemical reaction networks with partially finite state space
VL - 143
ER -
TY - JOUR
AB - Based on the measurements of noise in gene expression performed during the past decade, it has become customary to think of gene regulation in terms of a two-state model, where the promoter of a gene can stochastically switch between an ON and an OFF state. As experiments are becoming increasingly precise and the deviations from the two-state model start to be observable, we ask about the experimental signatures of complex multistate promoters, as well as the functional consequences of this additional complexity. In detail, we i), extend the calculations for noise in gene expression to promoters described by state transition diagrams with multiple states, ii), systematically compute the experimentally accessible noise characteristics for these complex promoters, and iii), use information theory to evaluate the channel capacities of complex promoter architectures and compare them with the baseline provided by the two-state model. We find that adding internal states to the promoter generically decreases channel capacity, except in certain cases, three of which (cooperativity, dual-role regulation, promoter cycling) we analyze in detail.
AU - Rieckh, Georg
AU - Tkacik, Gasper
ID - 2231
IS - 5
JF - Biophysical Journal
SN - 00063495
TI - Noise and information transmission in promoters with multiple internal states
VL - 106
ER -
TY - JOUR
AB - Maximum entropy models are the least structured probability distributions that exactly reproduce a chosen set of statistics measured in an interacting network. Here we use this principle to construct probabilistic models which describe the correlated spiking activity of populations of up to 120 neurons in the salamander retina as it responds to natural movies. Already in groups as small as 10 neurons, interactions between spikes can no longer be regarded as small perturbations in an otherwise independent system; for 40 or more neurons pairwise interactions need to be supplemented by a global interaction that controls the distribution of synchrony in the population. Here we show that such “K-pairwise” models—being systematic extensions of the previously used pairwise Ising models—provide an excellent account of the data. We explore the properties of the neural vocabulary by: 1) estimating its entropy, which constrains the population's capacity to represent visual information; 2) classifying activity patterns into a small set of metastable collective modes; 3) showing that the neural codeword ensembles are extremely inhomogenous; 4) demonstrating that the state of individual neurons is highly predictable from the rest of the population, allowing the capacity for error correction.
AU - Tkacik, Gasper
AU - Marre, Olivier
AU - Amodei, Dario
AU - Schneidman, Elad
AU - Bialek, William
AU - Berry, Michael
ID - 2257
IS - 1
JF - PLoS Computational Biology
SN - 1553734X
TI - Searching for collective behavior in a large network of sensory neurons
VL - 10
ER -