Math Insight

Short-term memory project

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Total points: 30

Short-term memory is the capacity to remember small amounts in an active state for a short amount of time. Unlike long-term memory, which is stored by changing the connections among neurons, a short-term memory is maintained through the activity of neurons in a fixed network.

One way to explore the neuronal basis of short-term memory is through measuring the activity of neurons while animals perform tasks that require short-term memory. In one version of such a task, monkeys attempt to remember the location of an object presented on a computer monitor. At the beginning of a trial, no objects appear on the monitor (we'll call it a blank screen). Then, an object is briefly projected on the monitor. During the subsequent delay period, the blank screen is shown again. Although no objects appear on the monitor so that the visual stimulus is identical to that from the beginning of the trial, the monkey must actively remember the object's location in order to report its location at the end of the delay period and receive a reward.

Neurons in an area of the brain called the prefrontal cortex have been observed to be involved in this short-term memory task. These neurons show low activity during the initial blank screen present before an object appears. However, depending on the location of the object that was projected, a neuron may fire at a higher rate during the delay period, even though the visual stimulus is identical to that from the beginning of the trial. This delay period activity of certain neurons appears to coincide with times when the monkey is remembering that the object was in a certain location.

Let's imagine that this delay period activity originates in the prefrontal cortex. (It's possible that the delay period activity originates in another brain region that then sends inputs to the prefrontal cortex, in which case our model would be a better fit to that other brain region.) If we imagine that the input to the prefrontal cortex is the same during the initial blank screen and the delay period, then the prefontal cortex must be able to sustain different patterns of activity in the presence of the same input. The history of input to the prefrontal cortex would determine which activity patten was activated. In other words, the prefrontal cortex would be exhibiting short-term memory.

In this project, we'll seek to build and analyze a simplified mathematical model that exhibits some of the properties of a short-term memory network. The overarching questions for this project are:

  • What types of neuronal circuitry could support short-term memory?
  • What are the mechanisms that could underlie short-term memory?

A neuronal network consists of a myriad of neurons interconnected through a web of connections. Rather than developing a model with thousands of differential equations that track each neuron, our goal will be to capture the influence of the network with a simple two-dimensional model that tracks the activity of just two neurons. (The model could also represent two groups of neurons, but we'll imagine they are individual neurons to keep the vocabulary simple.)

We will represent the activity of each neuron by its firing rate. The output of each neuron is actually a sequence of spikes (called action potentials), and we'll model the rate at which each neuron is emitting these spikes. The model for the dynamics of these rates should capture how the input to a neuron increases the neuron's firing rate. With such a simple model, we can then analyze the influence of the connectivity between the neurons to see how we can adjust the connectivity to create short-term memory in the network.

  1. Step 1: map biology to math
    1. Since we are modeling the neurons' activity by just their firing rate, the first component of the single neuron model is its $f-I$ curve, the curve that gives its output frequency $f$ as a function of the input $I$. We'll assume the two neurons have the same $f-I$ curve given by the function $$S(I) = \frac{M I^2}{\sigma^2 + I^2}$$ for $I \ge 0$. (We'd assume that $S(I)=0$ for $I<0$, but we can ignore this detail since we won't have negative input in this model.) Set the parameters $M=100$ and $\sigma=20$. $S$ gives the output in spikes per second as a function of the input $I$.

      Plot the function $S(I)$ for $0 \le I \le 200$. Make the following observations about the $f-I$ curve.

      1. If the neuron receive no input, what is its firing rate?
      2. If the neuron receives a huge amount of input, how fast does it fire? What is its maximum firing rate? This maximum firing rate corresponds to which parameter of $S$?
      3. If the neuron receives an input of $I=\sigma = 20$, what is its firing rate? How does that compare to the maximum firing rate?
    2. Let $R_1$ be the firing rate of neuron 1 and $R_2$ be the firing rate of neuron 2. If neuron 1 receives the input $I_1$ and neuron 2 receives the input $I_2$, model the dynamics of the firing rate by \begin{align*} \diff{R_1}{t} &= \frac{1}{\tau}(-R_1 + S(I_1))\\ \diff{R_2}{t} &= \frac{1}{\tau}(-R_2 + S(I_2)) \end{align*} where the time constant $\tau$ is 10 ms (milliseconds). If $I_1$ and $I_2$ are fixed numbers, these equations are uncoupled. Let's examine the equation for $R_1$ in this case.

      Set $I_1=20$, so that $S(I_1)$ is a number (determined above). What is the equilibrium of the $R_1$ equation viewed as a single one-dimensional differential equation? Is the equilibrium stable or unstable?

      Your result shouldn't depend on the value of $I_1$. If $I_1$ is a fixed number, then $R_1$ should move toward $S(I_1)$. Given an input $I_1$, $R_1$ approaches the value of its $f-I$ curve, $S(I_1)$. (How quickly it approaches this value depends on the time constant $\tau$.) The same thing is true for $R_2$; if its input is fixed to $I_2$, it will approach the value $S(I_2)$.

    3. To model a network, we want the input of neuron 1 to depend on the firing rate of neuron 2 and vice versa. The strength of this interaction will be captured by a coupling strength parameter $W$. Let the input to neuron 1 be $W$ times the firing rate of neuron 2: $I_1 = WR_2$. Similarly, let $I_2 = WR_1$. We say that neuron 1 is connected to neuron 2 with coupling strength $W$ and neuron 2 is connected to neuron 1 with the same coupling strength. Our network is symmetric.

      Write down the system of equations for $R_1$ and $R_2$ using this substitution for $I_1$ and $I_2$. Write the network in two ways: (1) in terms of the function $S(I)$, as in the previous part, and (2) substituting the definition of $S(I)$ to obtain a system of differential equations for $R_1$ and $R_2$ with parameters $W$, $M$, $\sigma$, and $\tau$. The first form looks simpler and is easier to interpret, so we'll usually prefer that form. The second form, though, gives all the details, and you might prefer it if you don't like having a mysterious function $S$ in your equation.

  2. Step 2: analyze the model

    For the model analysis, we let $\tau=10$, $M=100$, and $\sigma=20$ for all cases. We will analyze the dynamics for three different values of $W$: $W=0.3$, $0.4$, and $0.5$.

    1. Let $W=0.3$. Derive the equations for the nullclines and draw the nullclines on the below phase plane.

      Hint: start with the $R_2$-nullcline, as it will just be a stretched or compressed version of the function $S$. The $R_1$-nullcline will be identical, only with the roles of $R_1$ and $R_2$ swapped.

      Find all equilibria and plot them on the phase plane.
      To find the equilibria, you can assume that $R_1=R_2$ (due to symmetry of the nullclines). Plug $R_1=R_2$ into one of the nullclines, and multiply through to get a cubic equation. That cubic equation should have one obvious solution that you can factor out, leaving a quadratic equation for potentially two more equilibria. Use the quadratic formula to find additional equilibria, but ignore any complex solutions, as complex equilibria don't make any sense.

      Classify all equilibria. (It may be simplest to write the Jacobian matrix in terms of the derivative of $S$ and calculate the $S'(I)$ separately.)

      Draw representative direction vectors in each region of the phase plane.

      Sketch two solutions consistent with the direction vectors and equilibria classifications. For the first solution, use the initial condition $(R_1,R_2)=(20,5)$; for the second, use the initial condition $(R_1,R_2)=(80,20)$.

    2. We won't analyze the case when $W=0.4$ thoroughly. Just draw the nullclines, calculate the equilibria, and draw the equilibria on the phase plane.

      When comparing the $W=0.4$ case to the previous $W=0.3$ case, you should notice that the stretching of the nullclines has changed just enough to make a qualitative difference in the phase plane picture.

    3. Analyze the case when $W=0.5$.

      Draw the nullclines, calculate the equilibria, and draw the equilibria on the phase plane.

      If you discover than an equilibrium is a saddle, then calculate the eigenvectors of the Jacobian matrix evaluated at the equilibrium. Draw a pair of arrows heading into the equilibrium corresponding to the eigenvector with the negative eigenvalue and label them “stable direction.” Draw a pair of arrows heading out of the equilibrium corresponding to the eigenvector with the positive eigenvalue and label them ”unstable direction.”

      Draw a direction arrow in each region of the phase plane and on each segment of the nullclines.

      Sketch solution trajectories with the initial conditions $(R_1,R_2)=(20,5)$ and $(R_1,R_2)=(80,20)$. In this case, you should get quite different end results for the two different initial conditions.

  3. Step 3: interpret the model analysis biologically

    In the model, for simplicity, we did not represent any inputs to the network. We could have added input terms to the argument of each $S$, can you could have analyzed phase planes corresponding to different levels of input. However, since we are most interested in the situation where there are no inputs (corresponding to the initial blank screen or the delay period in the monkey experiments), we can probe the short-term memory behavior of the network without explicitly including inputs in the model.

    To explore the short-memory behavior, imagine the following. Before the initial blank screen at the beginning of the trial, the activity of the network is set to a low state (using a mechanism that we aren't including in our model). In terms of our model, the initial blank screen state corresponds to a low initial condition, such as the initial condition $(R_1(0),R_2(0)) = (20,5)$ that you analyzed or even smaller values of $R_1(0)$ and $R_2(0)$.

    During the presentation of the object, the network receives input large enough to driving the firing rates to moderately high values. Since we didn't include these inputs in our model, we will just imagine that, when the delay period begins, the firing rates start with moderately high values. The delay period corresponds to moderately high initial conditions such as the initial condition $(R_1(0),R_2(0)) = (80,20)$ that you analyzed, though $R_1(0)$ and $R_2(0)$ could be lower or higher.

    We also did not include any adaptation or other mechanism that would allow a memory to degrade. Hence, although we are modeling short term memory, which typically lasts for seconds, any “memory” in this model will last forever. Given the small time constant $\tau=10$ ms, the transient behavior of the model with last only a short time, much shorter than the few seconds that we want the memory to last. Hence, the short-term memory we would like to create will be represented by the long-term dynamics of the model!

    1. When $W=0.3$, what happens to $R_1(t)$ and $R_2(t)$ after a long time, independent of the initial condition? Justify your answer based on the results of your phase-plane analysis.

      Could the network represent a short-term memory? In other words, is it possible to have small initial conditions (the initial blank screen) lead to low activity in the long term and moderately large initial conditions (the beginning of the delay period) lead to higher activity in the long term? (Again, remember that the long-term dynamics of the model represent the short-term memory!)

    2. When $W=0.5$, what happens to $R_1(t)$ and $R_2(t)$ after a long time? How many possible long-term states are there for $(R_1(t),R_2(t))$? If there are more than one, how does the resulting long-term state depend on the initial conditions? Justify your answer based on the results of your phase-plane analysis.

      Could the network represent a short-term memory? If so, explain how one could determine from the neuronal activity if monkey was looking at the initial blank screen or if the monkey was remembering during the delay period that an object had been previously presented in a particular location?

    3. An intermediate state between the coupling strength $W=0.3$ and the coupling strength $W=0.5$, is the special case of $W=0.4$. By comparing the nullclines from the three cases, explain what critical change happens in the nullclines right at $W=0.4$. What happens to the number of stable and unstable equilibria as you go from $W < 0.4$ to $W > 0.4$? (Don't worry about the stability right at $W=0.4$.) You don't need to do additional analysis, but can just assume that $W=0.3$ captures the important features for $W < 0.4$ and $W=0.5$ captures the important features for $W > 0.4$.

      To have a short-term memory in this network, how many stable or unstable equilibria must you have? (If you need two different possibilities for long-term behavior, what must be true about the equilibria?) What can you conclude must be true about the coupling between the neurons in order for the network to support a short-term memory?

    4. Given the coupling of the network, if the firing rate $R_1$ of neuron 1 increases, what happens to the input of neuron 2? How would that input change alter the firing rate, $R_2$, of neuron 2? How would that firing rate change alter the input to neuron 1? And, to close the loop, how could that influence the firing rate $R_1$ of neuron 1? What type of feedback loop is present in the network? (Positive feedback or negative feedback?)

      When $W > 0.4$, the saddle (and initial conditions along its stable direction) represents the point right where this feedback loop can change the direction of the dynamics. If the system starts at an initial condition near the saddle, but upward and to the right of its stable directions, describe what happens to the dynamics of the solution $(R_1(t),R_2(t))$? Was the feedback strong enough to radically change the behavior of the system (compared to what would have happened if $W < 0.4$)?

      If the system starts at an initial condition near the saddle, but downward and to the left of its stable directions, describe what happens to the dynamics of the solution $(R_1(t),R_2(t))$? Was the feedback strong enough to radically change the behavior of the system (compared to what would have happened if $W < 0.4$)?

    5. As $W$ increases, the saddle point moves downward and to the left (you can take that as a fact). As $W$ increases, what happens to the basin of attraction of the memory state (i.e., the region of initial conditions that are attracted to the stable equilibrium with high firing rate)?

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Math 2241, Spring 2016