A mathematical framework for inferring connectivity in probabilistic neuronal networks
Introduction
In many cases where one wishes to infer the connectivity among nodes in a network, only a small fraction of the nodes can be sampled simultaneously. For example, neuroscientists can measure the individual activity of only tens, or possibly hundreds, of neurons simultaneously, despite the fact that the neural networks of even small areas of the brain contain millions of neurons. Similar situations exist when studying, for example, gene networks, communication networks, or social networks. The recurring feature is the presence of large numbers of unmeasured nodes. The effects of connections involving these unmeasured nodes can confound attempts to infer the “subnetwork” of connections among the measured nodes. For example, connections between a single unmeasured node and multiple measured nodes could cause the measured nodes to appear connected even if no direct connection among the measured nodes exists. The focus of this work is a mathematical framework that can, under suitable assumptions, provide a method to control for the effects of unmeasured nodes.
We restrict our attention to networks that can be represented as directed graphs (digraphs), where the direction of a connection indicates a causal connection from one node to another. We view each node as having some measurable “activity” (we will be more specific below). A causal connection means the activity in one node affects the future activity of another node (but not the reverse, unless there also happens to exist a reciprocal connection). We can hence refine our goal to inferring the causal subnetwork of connections among the measured nodes (see Fig. 1). We wish to identify the causal influences among measured nodes, distinguishing such causal network connections from those connections involving unmeasured nodes where no such causal influence is present. For example, in the case where a single unmeasured node has connections onto multiple measured nodes (a “common input” configuration, illustrated with dashed thick lines in Fig. 1), there is no causal connection among the measured nodes. Hence, to reconstruct the causal subnetwork among measured nodes, we must control for the possibility of common input connections from unmeasured nodes.
If measured node 1 has a causal connection onto measured node 2 (as in Fig. 1), one would expect the connection to introduce correlations into the activity of the nodes. The activity of node 2 would be correlated with a delayed version of the activity of node 1, where the delay corresponds to the time required for the influence of node 1’s activity to affect node 2. However, common input connections from an unmeasured node could induce similar correlation between the measured nodes’ activity. Consider the example from Fig. 1 where an unmeasured node has a connection onto measured nodes 3 and 4. If the connection onto node 4 has a shorter delay than the connection onto node 3, then the activity of node 3 will be correlated with a delayed version of the activity of node 4. From observing this correlation in the activity of nodes 3 and 4, one might naively and incorrectly conclude that node 3 received a causal connection from node 4.
This paper describes a method that can distinguish the causal subnetwork from common input connections, subject to a certain form of ambiguity in the identity of nodes that we call “subpopulation” ambiguity. We argue that, at least in a large number of neuroscience experiments, this subpopulation ambiguity is already present in how nodes are identified. Hence, even without the presence of unmeasured nodes, a determination of connectivity will be subject to this subpopulation ambiguity. When our analysis can be applied, the presence of unmeasured nodes adds little additional ambiguity to the determination of connectivity.
This research is motivated by neuroscience applications, and this initial formulation is designed to be immediately applicable to neuroscience experiments. For clarity and to emphasize that some elements of the analysis are specialized to neuroscience, we will primarily use the language of neuroscience, referring to the network nodes as neurons. Nonetheless, we believe the basic approach can be generalized to have wider application.
In Section 2, we detail the subpopulation ambiguity that is present in our results. In Section 3, we describe the model framework and the assumptions of the analysis. We present the analysis of the model in Section 4 and derive our estimates of causal network structure. We demonstrate the results applied to simulations in Section 5 and discuss the results in Section 6.
Section snippets
The subpopulation ambiguity
The definition of a subpopulation of neurons is based on the relationship between neural activity and any measured external variables. (A group of neurons whose activity has a similar relationship to the external variables will be considered part of the same subpopulation.) We first describe the external variables before discussing the subpopulation ambiguity.
The model network
Our results are based on a fairly generic class of probabilistic causal network models in discrete time. Since the measured activity of a neuron is the sequence of spike times, we model the activity as a point process. Rather than introduce standard point process notation (see, e.g., Refs. [1], [2]), we jump immediately to the formulation in discrete time, which is all we need for our analysis.
Initially, imagine that we have discretized time sufficiently finely so that a neuron can have at most
Analysis of model network
We seek to develop a method to determine the connectivity among measured neurons under the assumption that the activity of all neurons was generated according to the network model (3.3). The presence of unmeasured neurons will prevent us from completely succeeding, as our connectivity estimates will be subject to the subpopulation ambiguity discussed in Section 2.
We divide the set of all neural indices into two non-overlapping sets: containing the indices of measured neurons and containing
Simulation results
To illustrate our approach, we simulated several small networks. We designated two neurons as measured neurons and recorded the spike times of only those two neurons. The remaining neurons were unmeasured, and we ignored their spike times in the analysis. From the spike times of the two measured neurons and the external variables, we attempted to determine the connectivity between the two measured neurons.
Discussion
We have developed a model-based modular approach to identifying causal connections in a neural network where many neurons remain unmeasured. The approach is modular because the analysis works with a large class of models (which determine the f (w, X; θ) of Eq. (3.3)). The network analysis can use models, and algorithms to determine their parameters, that have been developed independently. When a model captures the neurons’ behavior sufficiently well, we can distinguish causal connections from
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This research was supported in part by the National Science Foundation Grant DMS-0415409.