ZFIN Figure: van der Plas et al., 2023, Figure 2

Figure 2

cRBM is optimized to accurately replicate data statistics.

(A) Schematic of the cRBM architecture, with neurons $v_{i}$ $v_{i}$ on the left, HUs $h_{μ}$ $h_{μ}$ on the right, connected by weights $w_{i, μ}$ $w_{i, μ}$ . (B) Schematic depicting how cRBMs generate new data. The HU activity $h (t)$ $h (t)$ is sampled from the visible unit (i.e. neuron) configuration $v (t)$ $v (t)$ , after which the new visible unit configuration $v (t + 1)$ $v (t + 1)$ is sampled and so forth. (C) cRBM-predicted and experimental mean neural activity $⟨ v_{i} ⟩$ $⟨ v_{i} ⟩$ were highly correlated (Pearson correlation $r_{P} = 0.91$ $r_{P} = 0.91$ , $P < 10^{- 307}$ $P < 10^{- 307}$ ) and had low error ( ${nRMSE}_{⟨ v_{i} ⟩} = 0.11$ ${nRMSE}_{⟨ v_{i} ⟩} = 0.11$ , normalized Root Mean Square Error, see Materials and methods - ‘Calculating the normalized Root Mean Square Error’ ). Data displayed as 2D probability density function (PDF), scaled logarithmically (base 10). (D) cRBM-predicted and experimental mean Hidden Unit (HU) activity $⟨ h_{μ} ⟩$ $⟨ h_{μ} ⟩$ also correlated very strongly ( $r_{P} = 0.93$ $r_{P} = 0.93$ , $P < 10^{- 86}$ $P < 10^{- 86}$ ) and had low ${nRMSE}_{⟨ h_{μ} ⟩} = 0.15$ ${nRMSE}_{⟨ h_{μ} ⟩} = 0.15$ (other details as in C) (E) cRBM-predicted and experimental average pairwise neuron-HU interactions $⟨ v_{i} h_{μ} ⟩$ $⟨ v_{i} h_{μ} ⟩$ correlated strongly ( $r_{P} = 0.74$ $r_{P} = 0.74$ , $P < 10^{- 307}$ $P < 10^{- 307}$ ) and had a low error ( ${nRMSE}_{⟨ v_{i} h_{μ} ⟩} = 0.09$ ${nRMSE}_{⟨ v_{i} h_{μ} ⟩} = 0.09$ ). (F) cRBM-predicted and experimental average pairwise neuron-neuron interactions $⟨ v_{i} v_{j} ⟩$ $⟨ v_{i} v_{j} ⟩$ correlated well ( $r_{P} = 0.58$ $r_{P} = 0.58$ , $P < 10^{- 307}$ $P < 10^{- 307}$ ) and had a low error ( ${nRMSE}_{⟨ v_{i} v_{j} ⟩} = - 0.09$ ${nRMSE}_{⟨ v_{i} v_{j} ⟩} = - 0.09$ , where the negative nRMSE value means that cRBM-predictions match the test data slightly better than the train data). Pairwise interactions were corrected for naive correlations due to their mean activity by subtracting $⟨ v_{i} ⟩ ⟨ v_{j} ⟩$ $⟨ v_{i} ⟩ ⟨ v_{j} ⟩$ . (G) cRBM-predicted and experimental average pairwise HU-HU interactions $⟨ h_{μ} h_{ν} ⟩$ $⟨ h_{μ} h_{ν} ⟩$ correlated strongly ( $r_{P} = 0.73$ $r_{P} = 0.73$ , $P < 10^{- 307}$ $P < 10^{- 307}$ ) and had a low error ( ${nRMSE}_{⟨ h_{μ} h_{ν} ⟩} = 0.17$ ${nRMSE}_{⟨ h_{μ} h_{ν} ⟩} = 0.17$ ). (H) The low-dimensional cRBM bottleneck reconstructs most neurons above chance level (purple), quantified by the normalized log-likelihood (nLLH) between neural test data v_i and the reconstruction after being transformed to HU activity (see Materials and methods - ‘Reconstruction quality’). Median normalized = ${nLLH}_{cRBM}$ ${nLLH}_{cRBM}$ 0.24. Reconstruction quality was also determined for a fully connected Generalized Linear Model (GLM) that attempted to reconstruct the activity of a neuron v_i using all other neurons $v_{- i}$ $v_{- i}$ (see Materials and methods - ‘Generalized Linear Model’). The distribution of 5000 randomly chosen neurons is shown (blue), with median ${nLLH}_{GLM} = 0.20$ ${nLLH}_{GLM} = 0.20$ . The cRBM distribution is stochastically greater than the GLM distribution (one-sided Mann Whitney U test, $P < 10^{- 42}$ $P < 10^{- 42}$ ). (I) cRBM (purple) had a sparse weight distribution, but exhibited a greater proportion of large weights $w_{i, μ}$ $w_{i, μ}$ than PCA (yellow), both for positive and negative weights, displayed in log-probability. (J) Distribution of above-threshold absolute weights $| w_{i, μ} |$ $| w_{i, μ} |$ per neuron v_i (dark purple), indicating that more neurons strongly connect to the cRBM hidden layer than expected by shuffling the weight matrix of the same cRBM (light purple). The threshold $Θ$ $Θ$ was set such that the expected number of above-threshold weights per neuron $E (# w_{i} > Θ) = 1$ $E (# w_{i} > Θ) = 1$ . (K) Corresponding distribution as in (J) for PCA (dark yellow) and its shuffled weight matrix (light yellow), indicating a predominance of small weights in PCA for most neurons v_i. All panels of this figure show the data statistics of the cRBM with parameters $M = 200$ $M = 200$ and $λ = 0.02$ $λ = 0.02$ (best choice after cross-validation, see Figure 2—figure supplement 1) of example fish #3, comparing the experimental test data test and model-generated data after cRBM training converged.

Expression Data

Expression Detail

Antibody Labeling

Phenotype Data

Phenotype Detail

Acknowledgments

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