44 research outputs found
Statistical physics of pairwise probability models
Statistical models for describing the probability distribution over the
states of biological systems are commonly used for dimensional reduction. Among
these models, pairwise models are very attractive in part because they can be
fit using a reasonable amount of data: knowledge of the means and correlations
between pairs of elements in the system is sufficient. Not surprisingly, then,
using pairwise models for studying neural data has been the focus of many
studies in recent years. In this paper, we describe how tools from statistical
physics can be employed for studying and using pairwise models. We build on our
previous work on the subject and study the relation between different methods
for fitting these models and evaluating their quality. In particular, using
data from simulated cortical networks we study how the quality of various
approximate methods for inferring the parameters in a pairwise model depends on
the time bin chosen for binning the data. We also study the effect of the size
of the time bin on the model quality itself, again using simulated data. We
show that using finer time bins increases the quality of the pairwise model. We
offer new ways of deriving the expressions reported in our previous work for
assessing the quality of pairwise models.Comment: 25 pages, 3 figure
Exact mean field inference in asymmetric kinetic Ising systems
We develop an elementary mean field approach for fully asymmetric kinetic
Ising models, which can be applied to a single instance of the problem. In the
case of the asymmetric SK model this method gives the exact values of the local
magnetizations and the exact relation between equal-time and time-delayed
correlations. It can also be used to solve efficiently the inverse problem,
i.e. determine the couplings and local fields from a set of patterns, also in
cases where the fields and couplings are time-dependent. This approach
generalizes some recent attempts to solve this dynamical inference problem,
which were valid in the limit of weak coupling. It provides the exact solution
to the problem also in strongly coupled problems. This mean field inference can
also be used as an efficient approximate method to infer the couplings and
fields in problems which are not infinite range, for instance in diluted
asymmetric spin glasses.Comment: 10 pages, 7 figure
Intrinsic limitations of inverse inference in the pairwise Ising spin glass
We analyze the limits inherent to the inverse reconstruction of a pairwise
Ising spin glass based on susceptibility propagation. We establish the
conditions under which the susceptibility propagation algorithm is able to
reconstruct the characteristics of the network given first- and second-order
local observables, evaluate eventual errors due to various types of noise in
the originally observed data, and discuss the scaling of the problem with the
number of degrees of freedom
How biased are maximum entropy models?
Maximum entropy models have become popular statistical models in neuroscience and other areas in biology, and can be useful tools for obtaining estimates of mutual information in biological systems. However, maximum entropy models fit to small data sets can be subject to sampling bias; i.e. the true entropy of the data can be severely underestimated. Here we study the sampling properties of estimates of the entropy obtained from maximum entropy models. We show that if the data is generated by a distribution that lies in the model class, the bias is equal to the number of parameters divided by twice the number of observations. However, in practice, the true distribution is usually outside the model class, and we show here that this misspecification can lead to much larger bias. We provide a perturbative approximation of the maximally expected bias when the true model is out of model class, and we illustrate our results using numerical simulations of an Ising model; i.e. the second-order maximum entropy distribution on binary data.
Beyond inverse Ising model: structure of the analytical solution for a class of inverse problems
I consider the problem of deriving couplings of a statistical model from
measured correlations, a task which generalizes the well-known inverse Ising
problem. After reminding that such problem can be mapped on the one of
expressing the entropy of a system as a function of its corresponding
observables, I show the conditions under which this can be done without
resorting to iterative algorithms. I find that inverse problems are local (the
inverse Fisher information is sparse) whenever the corresponding models have a
factorized form, and the entropy can be split in a sum of small cluster
contributions. I illustrate these ideas through two examples (the Ising model
on a tree and the one-dimensional periodic chain with arbitrary order
interaction) and support the results with numerical simulations. The extension
of these methods to more general scenarios is finally discussed.Comment: 15 pages, 6 figure
Dynamics and Performance of Susceptibility Propagation on Synthetic Data
We study the performance and convergence properties of the Susceptibility
Propagation (SusP) algorithm for solving the Inverse Ising problem. We first
study how the temperature parameter (T) in a Sherrington-Kirkpatrick model
generating the data influences the performance and convergence of the
algorithm. We find that at the high temperature regime (T>4), the algorithm
performs well and its quality is only limited by the quality of the supplied
data. In the low temperature regime (T<4), we find that the algorithm typically
does not converge, yielding diverging values for the couplings. However, we
show that by stopping the algorithm at the right time before divergence becomes
serious, good reconstruction can be achieved down to T~2. We then show that
dense connectivity, loopiness of the connectivity, and high absolute
magnetization all have deteriorating effects on the performance of the
algorithm. When absolute magnetization is high, we show that other methods can
be work better than SusP. Finally, we show that for neural data with high
absolute magnetization, SusP performs less well than TAP inversion.Comment: 9 pages, 7 figure
Inferring network connectivity using kinetic Ising models
Poster presentation</p
Stimulus-dependent maximum entropy models of neural population codes
Neural populations encode information about their stimulus in a collective
fashion, by joint activity patterns of spiking and silence. A full account of
this mapping from stimulus to neural activity is given by the conditional
probability distribution over neural codewords given the sensory input. To be
able to infer a model for this distribution from large-scale neural recordings,
we introduce a stimulus-dependent maximum entropy (SDME) model---a minimal
extension of the canonical linear-nonlinear model of a single neuron, to a
pairwise-coupled neural population. The model is able to capture the
single-cell response properties as well as the correlations in neural spiking
due to shared stimulus and due to effective neuron-to-neuron connections. Here
we show that in a population of 100 retinal ganglion cells in the salamander
retina responding to temporal white-noise stimuli, dependencies between cells
play an important encoding role. As a result, the SDME model gives a more
accurate account of single cell responses and in particular outperforms
uncoupled models in reproducing the distributions of codewords emitted in
response to a stimulus. We show how the SDME model, in conjunction with static
maximum entropy models of population vocabulary, can be used to estimate
information-theoretic quantities like surprise and information transmission in
a neural population.Comment: 11 pages, 7 figure