Decisions constrained on Manifolds, Advanced Calculus in Neuroscience

Today I want to discuss how differential equations on manifolds are used in neuroscience. In a new article in nature neuroscience called Optimal Policy for Multi-Alternative Decisions by Dr Satohiro Tajima, Dr. Jan Drugowisch, Dr. Nisheet Patel, and Dr. Alexandre Pouget, they discuss a simplified model of decision networks that captures a wide variety of observed behaviors when animals decide between a group of N choices. Specifically, this model captures  IIA violations which refers to trial-to-trial variability in the decision making process. IIA violations occur because the animal is learning how best to perform the experimental task, and this learning process can affect the animal’s decision, making it decide slower or incorrectly deciding as the animal tries different solutions to experimental task. Their model also considers other well established phenomena, including urgency signals (the animal will be pressed to decide the longer it waits), and divisive normalization (The overall activation of all neurons is normalized by the other neurons). However, the coolest part of this paper is it uses a dynamical system that constrains all the useful information on a manifold. Controlling this manifold in time, allows them to explain all these different phenomena concisely in a simple model. 

To begin, a standard race model assumes that each choice is represented by an accumulating neuron that integrates evidence over time. As soon as one of these accumulators passes some fixed threshold the system has decided.  The novel insight in their model is using an optimal policy that is constructed using both a divisive normalization procedure. Here the output of all neurons $x_i$ is fed into a function $C_t = \frac{x_i}{K+\sum x_i}$ where $K$

is some constant. This is the direct definition of divisive normalization. It sums up the strength of all the signals and divides all neurons activities by this inhibitory input. 

Figure 2 is from the original paper. Figure 2A shows full 3-dimensional system. Here the 3 different thresholds are in red, green, and blue. The Black triangle is the underlying manifold that all solutions will be projected onto. Figure 2B Shows the dynamics on the triangular manifold. As time increases the 3 thresholds get closer and closer together. Figure 2C shows how the 3 accumulators, the linear urgency signal and the divisive normalization inhibition circuits are constructed. Figure 2D shows how the thresholds are changed as time increases. They are normally the faces of a cube, but as the urgency signal grows the intersection with the 2d manifold pushes the thresholds closer and closer to a central point on the manifold, until eventually a decision is made. 

However more realistic models require an urgency signal. This is a simple ramp current, so as time increases all neurons are receiving a larger excitatory signal that linearly ramps up proportional to time. Looking at the image from Figure 2 of the paper can give an insight of how this works. Interestingly this has a result of pushing all accumulating neurons  closer and closer to threshold, and eventually it will be forced to make a decision.   

Perhaps the most interesting part of this model however, is its use of manifolds. Here they give the example in a three choice system. Because of the symmetry, the line connecting 0,0,0 and 1,1,1 contains completely useless information. Therefore signal components along this vector can be ignored. Thus the N-dimensional system can be reduced to a N-1-dimentional system. That surface in the 3-D example is the equilateral triangle. What's more this surface is forced outward towards the point 1,1,1 by the urgency signal. This has the results of pushing the threshold projection onto the manifold inward. Thus, making the decision more likely to happen. 

Again by observing figure 2b we can see the reduced manifold surface. The red, blue, and green lines are the thresholds for decision, 1,2 and 3 respectively. All useful information in the signal can be projected and constrained on the triangle in figure 2b. This is the N-1-dimensional manifold where the decision will take place. As the signal undergoes a random walk on this triangle the urgency signal pushes the red, blue, and green thresholds inward. Thus it is increasingly likely a decision will be made. 

It turns out this simple model unites a large number of observed phenomena in decision making, and underlying it all is manifold that changes in time. 

Author: Alexander White


Source: Satohiro T., Jan D., Nisheet P., and Alexandre P.. Optimal policy for multi-alternative decisions, nature neuroscience(2019)
https://www.nature.com/articles/s41593-019-0453-9