Direct dependencies between neurons explain activity

Abstract

Our understanding of neural computation is founded on the assumption that neurons fire in response to a linear summation of inputs. Yet experiments demonstrate that some neurons are capable of complex functions that require interactions between inputs. Here we show, across multiple brain regions and species, that direct dependencies (without interactions between inputs) explain most of the variability in neuronal activity. Neurons are quantitatively described by models that capture the measured dependence on each input individually, but assume nothing about combinations of inputs. These minimal models, which are equivalent to logistic artificial neurons, predict complex higher-order dependencies and recover known features of synaptic connectivity. The inferred neural network is sparse, indicating a highly redundant neural code that is robust to perturbations. These results suggest that, despite intricate biophysical details, most neurons are described by simple artificial models.

Figures (16)

• Figure 1: Fig. \ref{['fig_maxEnt']}$|$ Quantifying the variability explained by direct dependencies.a, Activity (dots) of an output neuron $y$ and $n$ inputs $x_1,\hdots,x_n$. Within a window of width $\Delta t$, each neuron binarizes into active ($y=1$) or silent ($y = 0$).Rieke-01Schneidman-01 To study activity on the fastest available timescale, $\Delta t$ is defined by the sampling interval of a given experiment. b, The full input-output dependence is defined by the probability of activity in response to each of the $2^n$ combinations of inputs (top). The simplest dependencies reflect the direct responses to each input individually, the number of which grows linearly with $n$ (bottom). c, The minimal model, which has maximum entropy consistent with these direct dependencies,Jaynes-01Cover-01 is equivalent to a logistic artificial neuron.Hertz-01d, Hierarchy of entropies, where the difference $S_\text{tot} - S_\text{dir}$ quantifies the amount of variability captured by direct dependencies, $S_\text{dir} - S_\text{true}$ the variability due to higher-order and time-delayed dependencies, and $S_\text{true}$ the latent variability that cannot be explained by the inputs.Schneidman-02e, Logical function with error probability $\epsilon$ and binary inputs that are drawn independently at random. f-g, Hierarchy of entropies versus error rate for the AND and OR functions (f) and the XOR function (g). For AND and OR (f), the functions are almost exactly captured by direct dependencies ($S_\text{dir} \approx S_\text{true}$), while for XOR (g), direct dependencies explain none of the output variability ($S_\text{dir} = S_\text{tot}$).
• Figure 1: Fig. S\ref{['fig_error']}$|$ Training and test errors.a-d, Ratio of test and training log-likelihoods $\ell_\text{test}/\ell$ versus the number of inputs $n$ normalized by $n^*$ for populations of neurons in the mouse hippocampus (a),Gauthier-01 mouse visual cortex during responses to natural images (b) and spontaneous activity (c),Stringer-01 and the brain of C. elegans (d).Dag-01 Dashed lines indicate the complete model with $n = n^*$. e-h, Ratio of test and training errors $\varepsilon_\text{test}/\varepsilon$ versus the number of inputs $n$ normalized by $n^*$ for the same data as a-d. Values in the hippocampus are averaged over all $N = 1485$ neurons in the population (a and e); values in the visual cortex are averaged over 100 randomly-selected neurons among a population of $N = 11,445$ (b, c, f, and g); and values in C. elegans are averaged over all $N = 128$ neurons in the population (d and h).
• Figure 2: Fig. \ref{['fig_neuron']}$|$ Identifying a minimal set of optimal inputs.a, Population of $N = 1485$ neurons in the mouse hippocampus recorded as the animal runs along a virtual track (Methods).Gauthier-01Meshulam-03 For a randomly-selected output neuron (circle), we illustrate the $n = 100$ optimal input neurons (blue) and an equal number of random inputs (red). b, Within a randomly-selected five-minute window, we plot the activity of the output neuron (top) and activity probabilities predicted by the maximum entropy model [Eq. (\ref{['eq_P']})] for increasing numbers of optimal inputs (left) and random inputs (right). c, Direct entropy $S_\text{dir}$ versus the number of inputs $n$ for optimal inputs (blue) and random inputs (red). d, Co-activity rates between the output and all other neurons versus those predicted by the independent model with no inputs; line indicates equality, and shaded region indicates experimental errors (two standard deviations). e, With $n^* = 350$ optimal inputs, the model correctly predicts all remaining co-activity rates, and thus all direct dependencies on other neurons in the population (Methods). f, With the same number of random inputs, the model fails to predict many correlations.
• Figure 2: Fig. S\ref{['fig_mean']}$|$ Average variability explained by direct dependencies.a, Direct entropy $S_\text{dir}$ normalized by total entropy $S_\text{tot}$ for $n$ inputs chosen optimally (blue) or randomly (red). Lines and shaded regions represent means and one-standard-deviation error bars across all $N = 1485$ hippocampal neurons.Gauthier-01Meshulam-03 Dashed line indicates the average minimal number of inputs $n^*$ needed to capture all the direct dependencies. b-d, Normalized model entropy $S_\text{dir}/S_\text{tot}$ versus number of inputs $n$ for 100 random output neurons within a population of $N = 11,445$ cells in the mouse visual cortex during responses to natural images (b) and spontaneous activity (c),Stringer-01 and for $N = 128$ neurons in the brain of C. elegans (d).Dag-01e-f, For the complete models in a-d, we compare the minimal number of inputs $n^*$ needed to capture all direct dependencies (e) and the normalized entropies $S_\text{dir}/S_\text{tot}$ (f). Values and error bars represent means and standard deviations across neurons.
• Figure 3: Fig. \ref{['fig_minimax']}$|$ Small numbers of direct dependencies explain activity across systems and species.a, Direct entropy $S_\text{dir}$ normalized by total entropy $S_\text{tot}$ for $n$ inputs chosen optimally (blue) or randomly (red). Lines and shaded regions represent medians and interquartile ranges across all $N = 1485$ hippocampal neurons in Fig. \ref{['fig_neuron']}.Gauthier-01Meshulam-03 Dashed line indicates the minimal number of inputs $n^*$ needed to capture all the direct dependencies for the median neuron; note that this number varies between neurons. b-d, Normalized model entropy $S_\text{dir}/S_\text{tot}$ versus number of inputs $n$ for 100 random output neurons within a population of $N = 11,445$ cells in the mouse visual cortex during responses to natural images (b) and spontaneous activity (c),Stringer-01 and for $N = 128$ neurons in the brain of C. elegans (d).Dag-01 See Methods for experimental details. e-f, For the complete models in a-d, we compare the minimal number of inputs $n^*$ needed to capture all direct dependencies (e) and the normalized entropies $S_\text{dir}/S_\text{tot}$ (f). Values and error bars represent medians and interquartile ranges across neurons.