Causal networks in simulated neural systems

Abstract

Neurons engage in causal interactions with one another and with the surrounding body and environment. Neural systems can therefore be analyzed in terms of causal networks, without assumptions about information processing, neural coding, and the like. Here, we review a series of studies analyzing causal networks in simulated neural systems using a combination of Granger causality analysis and graph theory. Analysis of a simple target-fixation model shows that causal networks provide intuitive representations of neural dynamics during behavior which can be validated by lesion experiments. Extension of the approach to a neurorobotic model of the hippocampus and surrounding areas identifies shifting causal pathways during learning of a spatial navigation task. Analysis of causal interactions at the population level in the model shows that behavioral learning is accompanied by selection of specific causal pathways-"causal cores"-from among large and variable repertoires of neuronal interactions. Finally, we argue that a causal network perspective may be useful for characterizing the complex neural dynamics underlying consciousness.

Fig. 1

Target fixation model. (a) The agent controls head-direction (H) and eye-direction (not shown) in order to move a gaze point (G) towards a target (T). (b) Neural network controller. Six input neurons are shown on the left: v-inputs reflect displacement of G from T; h-inputs and e-inputs reflect proprioceptive signals (see text). Four output neurons are shown on the right: H-outputs control head velocity and E-outputs control the velocity of E relative to H. For clarity only 4 of the remaining 22 neurons and only a subset of the 256 connections are shown.

Fig. 2

(a) Causal connectivity in the target fixation model. Each panel shows input (v: v-inputs, e: e-inputs, h: h-inputs) and output neurons (E: E-outputs, H: H-outputs). Red arrows show input→output causality, green arrows show output→input causality, and blue arrows show reciprocal causality. Arrow width reflects magnitude of causal influence. (b) Effect of environment on causal connectivity. See text for details; color online

Fig. 3

Post-lesion fitness following lesions to INs as a proportion of the fitness of the intact network, plotted against (a) mean unit causal density of the IN and (b) mean absolute covariance of the IN with the remainder of the network (see text). Each panel shows Pearson’s correlation coefficient (r) as well as the corresponding p-value.

Fig. 4

Schematic of Darwin X’s simulated nervous system. There were two visual input streams responding to the color (IT), and width (Pr), of visual landmarks on the walls, as well as one odometric input signalling Darwin X’s heading (ATN). These inputs were reciprocally connected with the hippocampus which included ‘entorhinal’ cortical areas EC_in and EC_out, ‘dentate gyrus’ DG, and the CA3 and CA1 hippocampal subfields. The number of simulated neuronal units in each area is indicated adjacent to each area. This figure is reprinted with permission from Seth and Edelman (2007)

Fig. 5

Causal connectivity patterns for a representative CA1 reference unit during the first trial (left) and the last trial (right). Grey arrows show unidirectional connections and black arrows show bidirectional connections. The width of each arrow (and size of arrowhead) reflect the magnitude of the causal interaction

Fig. 6

Distinguishing causal interactions in neuronal populations. (a) Select a neural reference (NR), i.e., the activity of a particular neuron (yellow) at a particular time (t). (b) The context network of the NR corresponds to the network of all coactive and connected precursors, assessed over a short time period. (c) Assess the Granger causality significance of each interaction in the context network, based on extended time-series of the activities of the corresponding neurons. Red arrows indicate causally significant interactions. (d, e) The causal core of the NR (red arrows) is defined as that subset of the context network that is causally significant for the activity of the corresponding neuron (i.e., excluding both non-causal interactions (black arrows) and ‘dead-end’ causal interactions such as 5→4, indicated in blue). Color online

Fig. 7

(a) The context network for a representative NR in Darwin X. The thickness of each line (and size of each arrowhead) is determined by the product of synaptic strength and the activity of the presynaptic neuron. (b) The corresponding Granger network. Line thickness here reflects magnitude of the corresponding causal interaction. (c) The corresponding causal core. Networks were visualized using the Pajek program (http://vlado.fmf.uni-lj.si/pub/networks/pajek/), which implements the Kamada-Kawai energy minimization algorithm. Color online

Fig. 8

(a) Size of context networks as a function of trial number during learning, in terms of number of edges (K), for 15 different CA1 neuronal units. (b) Sizes of the corresponding Granger networks. (c) Sizes of the corresponding causal cores. Insets of panels (b) and (c) show the same data on a magnified scale. Reprinted with permission from Seth and Edelman (2007)

Fig. 9

Example simple networks (top row) and corresponding causal connectivity patterns (bottom row). (a) Fully connected network. (b) Fully disconnected network. (c) Randomly connected network. Grey arrows show unidirectional connections and black arrows show bidirectional connections. The width of each arrow (and size of arrowhead) reflect the magnitude of the causal interaction. Corresponding values of causal density (c_d) are also given