Time-warp-invariant computation with action potentials: Deductions about the Hopfield-Brody Mouse

by David MacKay, Sanjoy Mahajan, Ed Ratzer, James Miskin, David Ward and Seb Wills

This is our entry to competition A, This document was written Wed Oct 18 2000.
[An MPEG(asf) movie of DJCM describing this theory with the aid of two pieces of pizza is also available.]
[A press release is also available.]

Deductions about the Hopfield-Brody Mouse

From figure 6, we concluded that a typical alpha or beta cell, starting from a resting state, must receive about 20 excitatory action potentials within about 20 ms if it is to reach threshold, assuming all action potentials produce current pulses of similar size.

Gamma cells are said to be similar except for their timeconstants which are shorter (6ms) than those of alpha and beta cells, and the peak currents are three times as large. Maybe 8 simultaneous excitatory action potentials are needed to get a gamma cell going. Since alpha cells, when firing, fire only once every 20 ms or so, gamma cells require a remarkable coincidence of excitatory inputs to fire them - perhaps some coordinated synchronization of the alpha cells is necessary for this to happen. Deduction: It seems plausible that an essential part of the recognition process is a synchronization of alpha cells, a bit like Herz-Hopfield earthquake networks.

If all synaptic strengths were as small as depicted in figure 6, there would never be any response to stimulation coming to area W from area A, since each alpha and beta cell receives only one synapse from area A (according to the anatomy of fig 3), and the firing rate is never high enough to give the required twenty spikes in 20ms. The maximum typical firing rate is about one every 8 ms. We thus deduce that the synaptic strength of the A to W connections must be big enough for a single action potential to cause the corresponding alpha or beta cell to fire. The similarities between figures 4 and 5 lead us to believe that every action potential coming from an A cell causes an action potential in its corresponding alpha cell. [The pure tone of 5c(i) is likely to have activated only a few cells in area A, so it is unlikely that many other cells in area W become active, so 5c(i) is, we guess, a pure response to area A alone; the differences in c(ii) and c(iii) must be produced by lateral stimulation of the given cell by other alpha and beta cells.] However, the timing of these spikes can be altered - by plus or minus about 10ms? - by lateral stimulation from alpha and beta cells; and additional spikes can also be produced by lateral stimulation too, though they occur in smallish numbers, so these extra spikes may not be important.

We have made another observation that may be relevant: In figure 1, we see that the burst of action potentials that signifies recognition of "one" is very regular, with an inter-spike interval that varies smoothly, increasing with time. The interspike interval is 20 ms in the earliest case and 40 ms in the latest case. This period is much longer than the integration time and refractory period of the gamma neuron, so the local periodicity must arise because of periodicity in the stimulation of the gamma neuron. It is likely that each spike in the gamma neuron is being produced because of synchronized action potentials arriving from essentially the same neurons as the previous spike, and all those neurons must be spiking at roughly the same frequency. We notice that in the case (1(b)) where the speaker said "one" more slowly, the burst of spikes is not only delayed to match the end of the word, but, also, the interspike interval is longer. The interspike interval thus encodes the time since the start of the word, or the speaking speed.

The gamma cell responds with a spike if a conjunction of about eight auditory features are present in the correct relative temporal positions. These auditory features are represented by synchronized firing of alpha cells. These cells that fire synchronously are cells which have the same firing rate as each other at the particular time when the end of word is expected.

Note that partial time-warp invariance results in this system. (See figure, which shows firing rates as a function of time-since-stimulus, for two alpha neurons) If two events, A and B, happen in a word, and if there are cells whose response decays with time at different rates as shown in the upper figure, and if the rates, and thus the inter-spike intervals, match at the "end of word" time, then these cells are capable of giving a synchronized input to the gamma cell at the right time. At other times, they will send unsynchronized spikes to the gamma cell, and it will not respond, since its time constant of integration is so short, and it requires many spikes for it to fire. Now, what if the word is spoken at a much faster rate (a factor of two faster is shown in the lower picture); well, the two cells will come into synchrony at a different, earlier, time, and as you can see from the picture, that's exactly the right time for the end of word, assuming a linear distortion of the word. We have confidence in this explanation, because it accounts perfectly for the property noted above, that slower-spoken words elicit slower-firing-rate bursts in the gamma cell.

The learning rule of the gamma cell must be this: when the learning signal comes, make positive synaptic connections to all cells that are firing synchronously.

The learning rule of the alpha cells must be this: if a pair of alpha cells both got their connections to a gamma cell increased by the above rule, then make lateral connections between them such that the two cells will tend to synchronize their firing. The beta cells connections should also be adjusted in the same sort of way, but I am not sure exactly what role the beta cells play in producing synchronization between cells with similar interspike interval.

I can't tell what the learning rule for inhibitory connections from beta cells to gamma cells should be.

Addendum added 6th November 2000

We first note that our initial submission failed to observe the detail that each alpha or beta cell is not driven by a single spiking A cell, but rather, by a current that decreases linearly with time in the same way as the A cell's firing rate. This detail makes life much easier for our theory: the response of a cell to a large input current is to fire periodically; and any other inputs to the cell will advance or retard the phase of this periodic firing.

We would like, second, to modify our theory for the rule that was used to set the connections. In our original deductions we suggested that any alpha cells that were firing at the same rate when the end of word arrived, and were thus potentially synchronizable, should have a positive connection between them, and positive connections to the gamma neuron. However, this idea, while it would probably work fine, and seems biologically reasonable, would not give rise to the simple approximately-periodic firing of the gamma cell observed in the experiments. A model that fits better is as follows: when the end of the training word arrives, find all alpha cells that are firing at some rate "r"; make positive connections between all pairs of these cells, and make positive connections from them to the gamma cell. The gamma cell will then respond periodically at that rate "r" when the end of the word arrives, if spoken at the original rate. For other (faster/slower) rates of speaking, it will respond faster and slower than that rate r.

We have a theory about the role of beta cells, based on the observation that the response of an alpha or beta cell as a function of time looks almost identical for all stimuli, once time-aligned. If there were no connections received by beta cell from alpha cells, and alpha-alpha connections were set as suggested above, then the response of an alpha cell to a cacophony of many sounds would be of slightly higher frequency than its response to a pure tone stimulus, because all the phase advances would have the same sign.

The effect of the beta connections in such a situation must be to retard the phase of the oscillator by as close to equal an amount as the phase advances supplied by the alpha connections. The simplest way (but perhaps this is not biologically plausible) is to have every alpha cell paired up with an identically tuned beta cell, and wherever there is a connection from alpha cell i to another cell j, there should be an equal-strength connection from the corresponding beta cell i to the same cell j. This arrangement would ensure that the average synaptic strength from any cell to any other is zero, if we integrate over the firing cycle of the recipient cell. Only when two cells i and j happen to be near to firing at almost the same time will the existence of the i->j connection have an effect, namely a slight advance of phase of j, if it is firing behind i, so that they are closer to synchrony. Let's call this idea alpha-beta-nulling so we can refer to it.

This model might be easiest to think about if each pair of matched alpha and beta cells fired in a synchronized arrangement, the alpha having its influence on other cells a moment before the beta cell. Thus we might hypothesize a positive alpha to beta connection within each matched pair, to bring about this synchronization; but it is not clear that such synchronization of matched alpha and beta cells is required for the alpha-beta-nulling effect to work.

Our theory still leaves open the question of whether there are any connections from alpha and beta cells to beta cells. Ideally, each beta cell should simply be firing at the same rate as its associated alpha cell, so we can't see any need for a beta cell to receive any inputs from alpha and beta cells, apart, perhaps, from its partner alpha cell, as suggested above. We think that the system would function OK if either there are no other inputs to a beta cell, or if the beta cell receives inputs identical to those received by its partner alpha cell.

But we're not sure, so we are investigating experimentally these three alternatives:

  1. Beta cells receive no connections at all
  2. Beta cells are driven only by their matched alpha partners
  3. Beta cells receive full connections under identical rules to the alpha cells.

David MacKay <mackay@mrao.cam.ac.uk>
This document was written Wed Oct 18 20:39:13 2000; Last modified: Wed Dec 13 18:28:36 2000