Fig 1.
Neuron model and population coding framework.
A. Framework schematic. A stimulus s from a probability distribution p(s) is encoded by the spiking responses of a population of ON (red) and OFF (blue) cells. We optimize the cells’ nonlinearities by maximizing the mutual information between stimulus and spiking response. B. Each cell is described by a binary response nonlinearity ν with a threshold θ and maximal firing rate νmax. During a coding window of fixed duration T the stimulus is constant and the spike count k is drawn from a Poisson distribution with a mean rate ν. C. When measuring coding efficiency using the mutual information between stimulus and spike count response, the neurons’ thresholds can be interpreted as quantiles of the original stimulus distribution, thus mapping an arbitrary stimulus distribution p(s) into a uniform distribution (four thresholds shown).
Fig 2.
Mutual information when constraining the expected spike count.
A. The mutual information between stimulus and response for any mixture of N ON and OFF cells is identical when constraining the expected spike count, R. B. The optimal threshold intervals for all possible mixtures of ON (red) and OFF (blue) cells in a population of N = 6 cells that achieve the same mutual information about a stimulus from an arbitrary distribution p(s). C. The optimal threshold intervals for the equal ON-OFF mixture in a population of N = 6 cells and different values of R (equivalently, noise); see also D. Top: low noise (RN → ∞); middle: intermediate noise (RN = 1); bottom: high noise (RN → 0). D. The optimal threshold intervals as a function of 1/RN. E. The mean spike count required to transmit the same information (see A) by populations with a different fraction of OFF cells (α), normalized by the mean spike count of the homogeneous population with α = 0. The different curves denote RN = {0.1, 1, 5, 100}.
Fig 3.
Optimal linear decoding of stimuli.
A. Framework schematic. A stimulus s from a probability distribution p(s) is encoded by the spiking responses of a population of ON (red) and OFF (blue) cells. We optimize the cells’ nonlinearities by minimizing the mean squared error (MSE) between the original stimulus s and the linearly reconstructed stimulus y from the spiking response. B. Minimizing the MSE between a stimulus s (black) and its linear estimate y (blue) by a population of (6) ON and OFF cells, in the absence of noise. We show the optimal weight w1 and the center of mass 〈s〉1 of the first threshold interval (red dashes). C,D. Any ON-OFF population can achieve the same error with the same set of optimal thresholds and weights but a different constant, w0. C. 6 ON cells (w0 < 0). D. 3 OFF and 3 ON cells (w0 = 0). E. The optimal thresholds equalize not the area under the stimulus density (as in the case of the mutual information), but the area under its one-third power (Eq 8). The optimal thresholds are shown for the Laplace distribution. F. The information maximizing thresholds partition the Laplace distribution into intervals that code for stimuli with higher likelihood of occurrence (bottom), while minimizing the MSE pushes thresholds to favor rarer stimuli near the tails of the distribution (top). Threshold distributions are the same as in E. G. The cumulative optimal thresholds (compare to E).
Fig 4.
Optimal linear decoding of stimuli with noise depends on the ON/OFF mixture.
A. The MSE as a function of the fraction of OFF cells in the population, α, for a different expected spike count, R. The MSE was normalized to the MSE for the homogeneous population of all ON cells. The MSE is shown for N = 100 cells and for the Laplace distribution. Symbols indicate the MSE values realized with the thresholds in B and C. B. The optimal thresholds for the homogeneous population (black) partition the Laplace stimulus distribution starting with a much larger first threshold than the mixed population with 2/3 OFF cells (blue) and 1/3 ON cells (red). C. The optimal thresholds for the Laplace distribution for a homogeneous population (black) and a mixed population with 2/3 OFF cells (blue) and 1/3 ON cells (red). In B and C, R = 1. Note the difference in the optimal threshold distribution between the mixed ON/OFF and the homogeneous ON population, especially for small x = i/N (logarithmic in blue vs. linear in black).
Fig 5.
The optimal ON/OFF mixture derived from the linear readout is tuned to asymmetries in the stimulus distribution.
A. The MSE as a function of the fraction of OFF cells (α) normalized to that for the homogeneous population of all ON cells (α = 0). The MSE is shown for an asymmetric Laplace distribution with varying negative to positive bias −/+, expected spike count R = 1 and N = 100 neurons. B. The optimal fraction of OFF cells as a function of stimulus bias of the asymmetric Laplace distribution and R = 1. C. The optimal thresholds for the ON-OFF mixtures (50%, 66% and 75%) in A that yield the lowest MSE, while varying negative to positive bias −/+ = {1, 2, 4}. D. Same as A but for an asymmetric Laplace distribution with a negative bias −/+ = 2 and varying R (equivalently, noise). E. The optimal fraction of OFF cells as a function of R for different stimulus bias of the asymmetric Laplace distribution. F. The optimal thresholds for the ON-OFF mixtures (84%, 70% and any) in D that yield the lowest MSE, while varying R = {0.02, 0.1, ∞}.
Table 1.
List of experimentally measured ON and OFF neuron numbers in different sensory systems.
Fig 6.
Deriving a distribution of stimulus intensities from experimentally measured thresholds.
A. Our efficient coding framework enables us to predict the optimal distribution of thresholds given a known stimulus distribution. By reversing our framework, we derive the stimulus distribution from a distribution of measured thresholds assuming optimal coding under the two optimality criteria. B. Log-log plot of the cumulative distribution of the inverse of thresholds from measured dose-response curves of the entire population of ORNs in the Drosophila larva olfactory system [37]. This is well described by a power law with exponent −0.42. C. The probability distribution of the inverse of optimal thresholds derived from the data in B. This is well described by a power law with exponent −0.58. D. Predicted distribution of concentrations across different odorants when assuming optimal coding by maximizing information or minimizing the MSE of the best linear decoder. This is well described by a power law with exponents −0.58 and −1.74, respectively. The proportionality constant is not relevant (see Methods).
Table 2.
Numerical results of the maximal mutual information for different configurations of ON and OFF cells allowing overlap, as well as enforcing overlap, for populations of up to ten cells.
The mutual information was computed numerically (in bits) upon finding the optimal threshold configuration where all possible overlap scenarios were considered, and compared to the analytically computed value for non-overlapping cells (Eq 1). The specified ON-OFF configurations denote the optimal sequence of ON (‘N’ symbol) and OFF (‘F’ symbol) thresholds. We consider three different noise levels denoted by R.
Fig 7.
Thresholds θi and intervals between thresholds ui for a population of 6 cells.
Top: a homogeneous population with 6 ON cells; bottom: a mixed population with 3 ON and 3 OFF cells.
Fig 8.
The mapping of the threshold indices from the homogeneous population with only ON cells (top) to the mixed population with ON and OFF cells (bottom).
Fig 9.
Determining the first thresholds for a mixed population of ON and OFF cells, as a function of X = fOFF/fON.
For a symmetric Laplace distribution p(s) = 1/2e−|s|.