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 - 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 - GEN AU - Friedlander, Tamar AU - Mayo, Avraham E. AU - Tlusty, Tsvi AU - Alon, Uri ID - 9718 TI - Supporting information text ER -