Reliable biological communication with realistic constraints

Communication in biological systems must deal with noise and metabolic or temporal constraints. We include these constraints into information theory to obtain the distributions of signal usage corresponding to a maximal rate of information transfer given any noise structure and any constraints. Generalized versions of the Boltzmann, Gaussian, or Poisson distributions are obtained for linear, quadratic and temporal constraints, respectively. These distributions are shown to imply that biological transformations must dedicate a larger output range to the more probable inputs and less to the outputs with higher noise and higher participation in the constraint. To show the general theory of reliable communication at work, we apply these results to biochemical and neuronal signaling. Noncooperative enzyme kinetics is shown to be suited for transfer of a high signal quality when the input distribution has a maximum at low concentrations while cooperative kinetics for near-Gaussian input statistics. Neuronal codes based on spike rates, spike times or bursts have to balance signal quality and cost-efficiency and at the network level imply sparseness and uncorrelation within the limits of noise, cost, and processing operations.

Figure 1

Simple case of communication from three input states to three output states. Maximum information transfer with a constraint in the cost implies a penalization of the noisy states (the first two) and of the costly states (the third one). These abstract states can represent, for example, enzyme concentration values, the different firing patterns of a neuron, or the patterns of network activity.

Figure 2

Relationship between input and output distributions and the input to output transformation for maximum information transfer, here illustrated for the transformation from the dimensionless substrate concentration $\left[S\right]∕k$ to the dimensionless reaction velocity $v$. In the simple case of no noise present and no cost constraint, the output distribution is a constant and the transformation is the integral of the input distribution. More output range is then dedicated to the more probable inputs, as illustrated by the two equal intervals of substrate concentration.

Figure 3

Substrate distributions for simple reaction kinetics maximize the information transfer. For Michaelis-Menten kinetics $(n=1)$ the distribution has a maximum at the lowest substrate concentration and a heavy tail. Increasing the cooperativity $n$ of the kinetics corresponds to distributions with a maximum closer to the constant $k$ in the reaction kinetics in Eq. (12).

Figure 4

Maximum information transfer for nonstationary substrate statistics. (a) Example of a substrate distribution that has a decreasing variance in time. (b) To keep the match between input statistics and kinetics that assures maximal transfer, the kinetics has to adapt when the change in substrate statistics. In this case the kinetics has to increase its degree of cooperativity.

Figure 5

Comparison of the theoretical prediction for maximum information transfer with a cost constraint in Eq. (25) (solid line) to the experimental spike rate distributions of visual cortex neurons. The spike rate (number of action potentials per second) was taken from. [9] Their recordings were obtained using extracellular electrodes in awake monkeys while they watch a monitor showing natural scenes. An exponential decay of the signal usage in (a) assures cost efficiency. Low signal usage at low spike rates in (b) assures signal quality. There is a good correspondence between theory and experiments for all spike rates, as shown in (c).