Dissecting cascade computational components in spiking neural networks

Abstract

Finding out the physical structure of neuronal circuits that governs neuronal responses is an important goal for brain research. With fast advances for large-scale recording techniques, identification of a neuronal circuit with multiple neurons and stages or layers becomes possible and highly demanding. Although methods for mapping the connection structure of circuits have been greatly developed in recent years, they are mostly limited to simple scenarios of a few neurons in a pairwise fashion; and dissecting dynamical circuits, particularly mapping out a complete functional circuit that converges to a single neuron, is still a challenging question. Here, we show that a recent method, termed spike-triggered non-negative matrix factorization (STNMF), can address these issues. By simulating different scenarios of spiking neural networks with various connections between neurons and stages, we demonstrate that STNMF is a persuasive method to dissect functional connections within a circuit. Using spiking activities recorded at neurons of the output layer, STNMF can obtain a complete circuit consisting of all cascade computational components of presynaptic neurons, as well as their spiking activities. For simulated simple and complex cells of the primary visual cortex, STNMF allows us to dissect the pathway of visual computation. Taken together, these results suggest that STNMF could provide a useful approach for investigating neuronal systems leveraging recorded functional neuronal activity.

Fig 1. Workflow of STNMF.

(A) Illustration of a minimal neural network model with four presynaptic RGCs and one postsynaptic LGN neuron. (B) Illustration of STNMF analysis. Averaging of the ensemble of spike-triggered stimulus images yields a single STA filter. STNMF reconstructs this ensemble by approximating it with a set of modules and a matrix of weights. One of the modules is strongly correlated to one of the spikes/images indicated by stronger (black lines) or weaker (gray lines) weights. (C) Illustration of STNMF weight analysis. Synaptic weights inferred by each column of the STNMF weight matrix, and spikes contributed by each presynaptic neuron inferred by each row of the matrix. (Di-iii) STNMF outputs. (i) Receptive field (RF) components of presynaptic neurons. Spatial filters (top) as subunit components of STNMF, and the corresponding temporal filters. (ii) Nonlinearity and synaptic weights from presynaptic neurons to the postsynaptic neuron. Ground-truth values of the model (green). The values computed from the weight matrix (red). Here weights were set as [2 1.5 1.8 1.3] for four neurons. (iii) STNMF separates the whole set of postsynaptic spikes into a subset of spikes contributed by each presynaptic neuron. Model spikes (gray) and inferred spikes (colored) of each presynaptic neuron. Correlation matrix of spike trains from model and STNMF (left). (Right) The matrix of mutual information (MI) carried by inferred spikes for each presynaptic neuron indicates that inferred spikes are similar to model spikes.

Fig 2. STNMF inference of shared presynaptic cells from different outputs of postsynaptic cells.

(A) (Left) Illustration of a 2-layer network with two output neurons. Layer 1 (L1) has six cells, where the cell 3 & 4 target to both cells in layer 2 (L2). (Right) Inferred presynaptic cells independently from both cells (L2–1 and L2–2) of layer 2. Receptive fields computed by STA for cell L2–1 and L2–2 and inferred L1 cells. (B) Spike trains generated from layer 1 model cells (gray) and inferred by STNMF (colored). (C) Correlation matrices of spike trains for each cell of layer 1 computed between model cells and inferred spikes from L2–1 cell (left) and L2–2 cell (right), as well as between inferred spikes (middle).

Fig 3. STNMF analysis of a 3-layer model.

(A) Illustration of a 3-layer network. Layer 1 (L1) has six cells, where cell 3 & 4 target to both cells in layer 2 (L2). The layer 3 cell receives two layer 2 cells. STNMF was applied to the layer 3 cell. (B) Inferred presynaptic cells from the layer 3 cell. (Top) STA of cell L3–1. (Bottom) STNMF subunits are RFs of inferred L1 cells. (C) (Left) Spike trains generated from layer 1 model cells 1–6 (gray) and inferred from layer 3 cell by STNMF (colored). (Right) Matrices of corresponding correlation of spike trains between model and STNMF inference. (D) (Left) Spike trains generated from layer 2 model cells 1–2 (gray) and combined spikes inferred from layer 3 cell by STNMF (cell 1: red, cell 2: blue). (Right) Correlation matrices of spike trains between model and STNMF inference.

Fig 4. STNMF analysis using natural image stimulus.

(A) Similar network model as in Fig 4 but with natural images instead of white noise as stimuli. The STA failed to obtain the RF of Layer 3 cell. (B) Modeled RFs of Layer cells (top) and the STNMF inferred results with 17 subunits (bottom). (C) Dot product matrix of the model RFs and STNMF subunits showing that the first subunits resemble the model cells.

Fig 5. Mixture of ON and OFF cell types identified by STNMF.

(A) Illustration of a neural network with ON and OFF cells. Similar to Fig 1, except that there are both ON and OFF presynaptic neurons. (B) ON and OFF cells are separated from STNMF. The RF of the postsynaptic neuron computed by STA (left). RFs of presynaptic neurons identified as STNMF subunits (top) with their corresponding temporal filters (bottom). (C) Presynaptic RFs computed by spikes inferred from STNMF. (D) Using STNMF weight matrix to classify spikes, the relationship among spikes, weights, and subunits is established, seen from (left) sum of specific weights of each subunit, and (right) sum all weights in each column of the weight matrix. (E) Spikes from the model and inferred by STNMF (left), and the corresponding matrices of spike correlation.

Fig 6. STNMF analysis of a V1 simple cell.

(A) Illustration of a model simple cell as a 3-layer neural network with ON and OFF cells. (B) ON and OFF cells are separated from STNMF. The RF of the simple cell computed by STA. RFs of layer 1 cells were identified as STNMF subunits with their corresponding temporal filters. (C) (Left) Spike trains generated from layer 1 model cells 1–6 (gray) and inferred from layer 3 cell by STNMF (colored). (Right) Matrices of corresponding correlation of spike trains between model and STNMF inference. (D) (Left) Spike trains generated from layer 2 model cells 1–2 (gray) and combined spikes inferred from layer 3 cell by STNMF (cell 1: red, cell 2: blue). (Right) correlation matrices of spike trains between model and STNMF inference.

Fig 7. STNMF analysis of a V1 complex cell.

(A) Illustration of a model complex cell as a 3-layer neural network with ON and OFF cells. There are eight cells in layer 1, of which the first four cells 1–4 have mixed ON and OFF types, and the second four cells 5–8 are similar with the same locations but with opposite polarity. Layer 2 cells are simple cells as in Fig 5. Layer 3 cell is a V1 complex cell. (B) The RF of the complex cell calculated by STA. 8 Subunits obtained by STNMF. (C) (Left) Spike trains generated from layer 1 model cells 1–8 (gray) and inferred from layer 3 cell by STNMF (colored). For each meaningful subunit, spikes are separated by minimal values of the weight matrix as OFF spikes, and minimal as ON spikes, respectively. Totally, there are 8 classified spike trains. (Right) Matrices of corresponding correlation of spike trains between model and STNMF inference. (D) Spatial and temporal filters obtained by STA analysis using classified spikes.