Assignment #3: Decision making in spiking networks
Due: April 27, 2012 Goals
Scientific: Understand the recurrent-excitation model of neural integration. Scientific: Understand how competition can instantiate neural “coin-flipping”. Scientific: Understand how the spiking activity in this simply-described neural model can underlie the animal’s behavior.
Introduction
In this assignment we will try to build a model that accounts for the behavior of monkeys in the reaction time version of the random dot motion discrimination task (i.e. the task from Roitman and Shadlen (2002), which we discussed in class). In particular we will try to qualitatively account for two facts: the monkey doesn’t always correctly determine the direction of motion of a noisy stimulus, particularly on difficult trials; and his latency to response (reaction time) on the most difficult trials is larger than on easier trials. To achieve this, we will be recreating the “dueling integrators” model of Wang (2002).
In this model, inspired by the neural data reported by Roitman and Shadlen (2002), two populations of spiking neurons integrate “sensory inputs” reflecting the strength of motion in two opposite directions. The populations are in
competition with each other, such that only one population can increase its activity beyond a certain level and thereby, in the abstraction, determine the behavior of the animal. We will build this network in two steps: first, a single network of recurrently excitatory neurons will be designed to “integrate” its input. Next, two of these integrating networks will be combined along with a mutually inhibitory population to enforce the winner-take-all behavior - the duel, as it were, waged with the firings of action potentials.
Part One: Integration
In the Wang (2002) model, there are four populations of neurons: inhibitory (Inh); excitatory responsive to stimulus A (ExcA); excitatory responsive to stimulus B (ExcB); excitatory responsive to neither stimulus. In this lab we will ignore the fourth of these altogether to save on simulation time, leaving us with only Inh, ExcA, and ExcB. For all three, use the same neuron counts that Wang uses (see the “Experimental Procedures” (ha) section at the end of the paper for these numbers and all other parameters that I don’t specify for you). But first we will build ExcA alone.
Excitatory synapses with large time constants (i.e. NMDA synapses) are important for the integration behavior of the model. For this reason, we need a model that can include two classes of excitatory synapses, each with their own time constant. We will use the multi-compartmental model ‘iaf_cond_alpha_mc’, placing the AMPA synapses on the somatic compartment and the NMDA
synapses on the proximal dendrite. To set a parameter for the soma in this model, use:
nest.SetStatus([e_neurons[k]], 'soma', {"C_m": c_m_e})
To make a connection to a particular compartment, use the following idiom:
# the “receptor_type” parameter specifies which synapse will be the target. To see the choices, use:
nest.GetDefaults(‘iaf_cond_alpha_mc’) # then to make the synapse:
nest.CopyModel("static_synapse", "mySynapseName", {"weight": 1.0, 'receptor_type': 1, 'delay': 0.5})
nest.ConvergentConnect([excA], [excA], model=" mySynapseName ")
We will effectively bring the proximal dendrite into the somatic compartment, as well as shut off the distal dendrite compartment, by making the conductance between the former two insanely large, and the conductance to the latter zero.
nest.SetStatus([e_neurons[k]], {'g_sp': 4000.}); # g_sp = conductance from soma to proximal; effectively make proximal dendrite part of the soma
nest.SetStatus([e_neurons[k]], {'g_pd': 0.}); # effectively turn off distal dendrite
There are a couple of adjustments we need to make to the synaptic weights in order to account for the fact that the equations governing them in Wang (2002) are somewhat different than the alpha-model synapse employed in
iaf_cond_alpha_mc. In particular, the shape of the EPSC in the NEST model is an alpha function with a peak normalized to 1nS. We’d like the total area of the synaptic response to be the same for our model as for Wang (2002), which has AMPA and GABA synapses described by a single exponential and NMDA described by a double exponential with separate rise and decay time constants. One can (but is not required to) demonstrate that for AMPA and GABA, the appropriate conversion is:
weightInNest = weightInWang/e;
and for NMDA:
There is one additional correction we must make for the NMDA synaptic weights, to account for the fact that in Wang (2002), these synapses have conductance dependent on the post-synaptic membrane potential. Here they won’t have that dependence at all; however, since we can empirically see that the membrane potentials in the model largely reside between -55 and -50mV, we may just get away with it. Thus, take his expression for the effect of the Mg-block, and compute its value at -52.5mV. That is, in his integration of the NMDA synapses he divides gNMDA by a term (1 + [Mg2+]*someFunction(V(t))), and you should divide your gNMDA by (1 + [Mg2+]*someFunction(-52.5)).
With those adjustments, incredibly, we should be able to use the weight parameters directly from this decade-old paper. Impossibly.
However, the model will not integrate immediately because these parameters are designed to produce integration in the face of incoming inhibition, which we have not yet included. Therefore, we need to turn down the recurrent excitation (both AMPA and NMDA) to a level where the network’s recurrent feedback can sustain its output. Create a parameter that scales both of these connection strengths by a multiplicative factor, and determine what value of this parameter generates integration.
You can quantify whether the model is performing accurate integration with whatever methods you deem most appropriate. The two important aspects of integration that you should assess quantitatively are: linear response to a
constant stimulus [i.e., plot of slope of firing rate response versus stimulus input strength, Deliverable], and no change in firing response when the stimulus input ceases [Deliverable]. There is probably no way you can get perfect integration because of strongly this model relies on precise tuning (but feel free to prove me wrong, for a million bonus points! Just kidding). You should be able to achieve something like what I show below (which you should then quantify as described above). I used independent 50Hz poisson inputs to the AMPA synapses of each neuron to drive the network.
Part Two: Competition
Now we’ll connect two of these integrators together with weak excitation and feedback inhibition. As above, the weights should be the parameters from the paper with the adjustments described. I found that it also helped to make the excitatory to inhibitory NMDA synapses about twice as strong as prescribed in the paper, presumably to help offset for the lack of input from all those excitatory neurons we aren’t modeling. You may also want or need to play with the
recurrent excitation scaling factor again, now that these neurons are receiving inhibition, so it should need to be turned up. Stimulate the network with a 1-second input as described in the paper (you need not have as long of a baseline
period). If the firing rate goes to an extreme value, just plot the relevant part of the graph, i.e. scale the y-axis to focus on the part with the competitive dynamics. As described in the paper, the fact that the neural populations integrate their inputs means that their activity will increase faster for a stronger stimulus input and therefore, on those trials, a decision (triggered by the firing rate of one population reaching a threshold level) will be reached more rapidly. The end result of such a process would be a shorter reaction time for the monkey. Thus, we can estimate the reaction time distribution by calculating the
threshold-crossing times. Choose a threshold by visually estimating the “point of no return” for the firing rates of a given population (i.e., there should be no trials in which both populations cross this threshold), and plot the threshold crossing times for coherence = 0, 4, 10, and 40%, using at least 10 trials for each, with standard deviations, as in Fig 5B of Wang (2002) [Deliverable].
Finally, plot a psychometric function showing the monkey’s choice behavior for various coherence levels, as in the other part of Figure 5B in Wang (2002)
[Deliverable]. Again use the coherence values of 0, 4, 10, and 40, with at least 10 trials each, you may want to use more if you have time. Your plot need not match that figure exactly, but should be at 50% for zero coherence, and should not reach 100% performance at 4% coherence. Fit the curve with a weibull function to estimate the threshold as Wang (2002) does.
Extensions
Examine the dependence of the single pool model’s ability to integrate on the time constant of the NMDA receptors. To do this, determine what the new synaptic weight should be if the synaptic time constant is lowered, such that the area under the EPSC is constant (i.e., at a shorter time constant, NMDA
synapses will need to get stronger to deliver the same amount of current). Try at least five new time constants shorter than the original 100ms, and characterize the ability of the network to integrate (using the same parameters you used in Part One above, and using the same metrics to quantify integration performance) [Deliverable].
Examine the robustness of the model by picking three parameters (whichever you think are most interesting or most likely to make the model fail) and determine how far they can go from the values given in the paper, as a percentage, before the behavior of the network breaks down in one way or
another (it is up to you to determine what constitutes a breakdown). Comment on how this analysis impacts your belief about whether a biological system could in fact implement this process.
Figures
1- Simple integration. The two curves represent average firing rate of the single integrating pool with two different levels of stimulus input, delivered over the period of 500 to 1500ms. Firing rate increases faster (with larger slope) for the larger input (red). After input ceases at 1500ms, there is some small relaxation of firing rates but the separate levels are
maintained.
2- Competition on multiple trials. A green and a red (population representing the true motion direction and the other direction, respectively) curve is plotted for each of 20 trials of the competitive network.
3- Psychometric function indicating the “choice” of the model for different coherence values. In some small number of trials, neither population exceeds the threshold by the end of the trial; these are excluded in this plot, but treated as RT=max for the following plot.
