nature neuroscience • volume 5 no 8 • august 2002 709

Temporal integration is relatively sim-ple—it is nothing more than the accu-mulation (or sum) over time of somesignal. This simple process, however,turns out to be a powerful form of com-putation that is believed to underlie alarge variety of seemingly unrelatedbehaviors and cognitive functions,including eye movement1, navigation2,3,short-term memory of continuous vari-ables4,5 and mental rotation6. It mighteven be critical in the ability of neuralcircuits to perform statistical infer-ences7, which is remarkable, consider-ing that statistical inference representsone of the most promising computa-tional theories of higher brain func-tions, such as perception, decisionmaking, motor control and high-levelreasoning8–10. Simple or not, under-standing how the nervous system per-forms integration has been one of themost frustrating problems in computa-tional neuroscience. In this issue,Koulakov et al.11 propose a new theo-retical model to explain how neural net-works might integrate signals.

A classic example of a neural integra-tor is a circuit contained in the post-subiculum of the rat. Neurons in thiscircuit encode the direction of the rat’shead in world-centered coordinates, act-ing much like an internal compass2. Thiscompass is updated after each movementof the head, even when the animal is beingmoved by the experimenter in completedarkness. This is particularly remarkablebecause, in darkness, the main source ofinformation regarding head movementsis the discharge of neurons in the semi-circular canals of the inner ear, whichrespond to head acceleration, not headdirection. Head direction, however, can

firing rate, inhibitory input causes adecrease, and, most importantly, withoutinput, the firing rate of the network staysconstant. This last fact is most important,because it allows the network to remem-ber the accumulated input signal.

The problem with this approach is that itrequires incredible fine-tuning of parame-ters. Trying to maintain a particular firingrate in a neural network is like trying to geta marble to sit still on a table (Fig. 1a). If the

Digitized neural networks: long-termstability from forgetful neuronsAlexandre Pouget and Peter Latham

Understanding how realistic networks integrate input signals over many seconds has eludedneuroscientists for decades. Koulakov and colleagues now propose a computational model toexplain how bistable neurons might allow a network to integrate incoming signals.

be recovered in two steps. First, headvelocity can be computed by summing(that is, integrating) head accelerationover time. Then, head direction isobtained by repeating this operation, but,this time, on head velocity. This doubleintegration is believed to be implement-ed by the circuitry linking the vestibularsystem to the postsubiculum.

As these examples illustrate, integrationis emerging as a basic computational oper-ation used by the brain, much like theswitching of transistors is a basic computa-tional operation used by digital computers.Unfortunately, despite numerous experi-mental and theoretical studies, its neuralbasis is still very poorly understood. Butwhat is so complicated about it? After all, itis particularly easy to design an algorithmthat integrates a signal over (discrete) time:on each time step, simply add the value ofthe input signal to the accumulated value,the latter being stored in memory (suchintegrators can easily be built from com-ponents bought for a few dollars at any elec-tronics store). Two operations are requiredfor this: addition and memory. The diffi-culty for the brain lies in the memory oper-ation: ‘remembering’ the summed value isdifficult to model in realistic neuronsbecause such neurons integrate signals overa short time period (milliseconds), where-as biologically relevant signals occur over alonger period (seconds). In other words,neurons tend to forget what happened tothem only 10 ms ago; given this intrinsicforgetfulness, it has been difficult to under-stand how networks of neurons rememberthings that happened seconds ago.

One solution that has been proposedby theorists is to build networks of neuronsthat have plenty of internal feedback1,12.This internal feedback allows the networkto maintain a particular firing rate, even inthe absence of input—much the same waya group of very forgetful people couldremember a phone number by continual-ly repeating it to each other. Such networksact as integrators in the following way: exci-tatory external input causes an increase in

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Alex Pouget is in the Department of Brain andCognitive Sciences, University of Rochester,Rochester, New York 14627, USA. Peter Lathamis in the Department of Neurobiology, UCLA,Los Angeles, California, 90095 USA.e-mail: alex@bcs.rochester.edu; pel@ucla.edu

Fig. 1. Storing values with a marble on a table.(a) The location of the marble can be used tostore any analog value (1.5 in this example).Moreover, if there is enough drag on the sur-face that the marble stops quickly in theabsence of any external force, the position ofthe marble codes for the integral of the forceacting on it, that is, it acts as an integrator offorce. (b) If the table is not flat, the marble willroll toward a local minimum, effectively forget-ting the initial position of the marble, that is,the initial stored value. The characteristic timeit takes to approach the minimum is the ‘for-getting’ time. For neurons, the forgetting timeis around 10 ms, much too short to integrateinput signals over seconds. Building a networkwith a long forgetting time—a nearly flat sur-face—out of such forgetful neurons is hard;this is the ‘fine tuning’ problem. (c) The solu-tion proposed by Koulakov et al. is to put pitsin the surface, so the marble does not rolleven if the surface is slightly warped.

Ivelisse Robles

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710 nature neuroscience • volume 5 no 8 • august 2002

table is perfectly flat, the marble can‘remember’ any value for arbitrarily longtimes—if you want it to remember thenumber ‘1.5’, for example, simply put themarble at position 1.5 (Fig. 1a). The slight-est slope will cause the marble to drift, how-ever, and thus forget its stored value (Fig. 1b), and even if the table is perfectlyflat, the system is still sensitive to noise, asthe slightest tremor will cause the marble towander aimlessly. (With a little more work,the marble can be turned into an integra-tor. But to do this, there needs to be enoughdrag on the marble so that it stops quicklyin the absence of external forces. In the high-drag regime, the marble does more thansimply remember where you put it: it inte-grates the forces that act on it. Again, how-ever, the table needs to be perfectly flat, orthe marble will drift; it also needs to be noisefree, or the marble will wander.)

In digital computers, these problems—fine tuning of parameters and sensitivity tonoise—are avoided thanks to two tricks.First, all numbers are discretized; they arerepresented as binary strings (for instance,64 bits). Although this discretization pre-vents computers from integrating analogvalues, 64-bit strings are sufficiently longfor most practical purposes. Second, eachbit is in a very stable state, that is, its valuecannot change because of the internal noisein the computer. This is because the volt-age difference between the 0 and 1 thatmake up the binary code is much largerthan the voltage fluctuations due to theelectronic noise, and much larger than anyapplied forces, like those caused by strayelectric fields. Once a bit is set, it does notchange unless the CPU tells it to change.Integration, then, becomes trivial: simplyadd or subtract numbers from a counter.Because neither noise nor external forcescause bits to spontaneously flip, we areguaranteed that the counter will faithfullyaccumulate all additions and subtractions.

Applied to our table example, thisapproach would be equivalent to carvingpits on the surface of the table (Fig. 1c).Provided the pits are deep enough, there isno need for fine tuning; the table could beslightly tilted without dire consequences.Furthermore, the sensitivity to noise wouldbe greatly reduced: the marble would beunlikely to jump from one position to thenext when the table shakes. Granted, themarble could only store a finite number ofvalues, but this is not a problem if we makethe pits closely spaced—just as it isn’t aproblem in digital computers that repre-sent numbers using 64 bits.

In essence, this is the solution proposedby Koulakov et al.11, but applied to a recur-

istic in at least four ways if it is to describeintegrators in mammalian networks. First,inhibitory neurons need to be included.So far, the model contains only excitatoryneurons, which forces it to operate in anunrealistically weakly coupled regime (all-excitatory networks with realistic couplingare unstable, in the sense that they canonly fire at high rates12). Second, a morerealistic ratio of AMPA to NMDA recep-tors needs to be incorporated. Two con-ductance ratios were used in their models,5 and infinity—both significantly higherthan the experimentally observed ratio of0.5–1 (ref. 14). Finally, the size of the net-work needs to be scaled up: in mammaliannetworks, each neuron receives ∼ 5000inputs, much larger than the 100–300inputs used in the Koulakov et al. model.All of these effects increase the effectivenoise, which would make it more likely forthe bistable neurons to spontaneouslyswitch states and thus wreak havoc withstability. Whether or not the model isrobust to these more realistic featuresneeds to be investigated numerically.

Experimentally, the proposed modelmakes a very strong prediction: in areaswhere neural integrators exist, neuronsshould be bistable. Besides being a cleversolution to a hard problem, the falsifiabili-ty of the model is one of its best features.We look forward to future experiments thateither confirm or deny bistability as themechanism underlying neural integrators.

1. Seung, H. Proc. Natl. Acad. Sci. USA 93,13339–13344 (1996).

2. Taube, J. S., Muller, R. U. & Ranck, J. B. J. J. Neurosci. 10, 420–435 (1990).

3. Zhang, K. J. Neurosci. 16, 2112–2126 (1996).

4. Droulez, J. & Berthoz, A. Proc. Natl. Acad. Sci.USA 88, 9653–9657 (1991).

5. Camperi, M. & Wang, X.-J. J. Comput.Neurosci. 5, 383–405 (1998).

6. Georgopoulos, A. P., Lurito, J. T., Petrides, M.,Schwartz, A. B. & Massey, J. T. Science 243,234–236 (1989).

7. Deneve, S., Latham, P. & Pouget, A. Nat.Neurosci. 4, 826–831 (2001).

8. Gold, J. I. & Shadlen, M. N. Trends Cogn. Sci.5, 10–16 (2001).

9. Knill, D. C. & Richards, W. Perception asBayesian Inference (Cambridge Univ. Press,New York, 1996).

10. Wolpert, D. M. & Ghahramani, Z. Nat.Neurosci. 3, 1212–1217 (2000).

11. Koulakov, A., Raghavachari, S., Kepecs, A. &Lisman, J. E. Nat. Neurosci. 5, 775–782 (2002).

12. Amit, J. & Brunel, N. Cereb. Cortex 7, 237–252(1997).

13. Eken, T. & Kiehn, O. Acta Physiol. Scand. 136,383–394 (1989).

14. Andrasfalvy, B. K. & Magee, J. C. J. Neurosci.21, 9151–9159 (2000).

rent network of spiking neurons. It is easyto build a recurrent network that is stableat any one particular firing rate, but stabil-ity at a single rate does not solve the prob-lem of integration. Koulakov et al., however,took advantage of an important fact: thestable firing rate in a recurrent networkdepends on the number of neurons in thenetwork. This is because the drive to eachneuron depends on the number of neuronspresynaptic to it. More neurons meansmore drive and a higher firing rate; fewerneurons means less drive and a lower firingrate. Importantly, for a fixed number ofneurons, the network is stable. This leadsto the following key observation: a networkcan have many stable firing rates if neuronscan be effectively removed or added, and ittakes a certain activation energy to do so.Koulakov et al. accomplished this by build-ing a network out of interconnected bistableneurons. If external input is applied to sucha network, it can change the relative num-bers of quiet and active neurons: excitato-ry input switches some quiet neurons to theactive state, increasing the effective numberof neurons and thus increasing the averagefiring rate; inhibitory input switches someactive neurons to the quiet state, decreas-ing the effective number of neurons and therate; and no input does nothing to the effec-tive number of neurons and thus nothingto the rate. The resulting network is a neur-al integrator that needs little fine tuning—in models, at least, network operation isrobust to parameter changes of as much as±20%. Just as important, it is stable withrespect to noise, as it takes large fluctuationsto cause the bistable neurons to switchstates. This stability is consistent with pre-vious models in which bistable neuronswere used in models of working memory5.

Bistability is a key aspect of theKoulakov et al. model11, so it is natural toask how it might come about in real neu-rons. It is easy to imagine in principle: allone needs is a voltage-dependent inwardcurrent that is activated when a neuron isfiring and inactivated when the neuron issilent. The existence of bistable neurons incortex is more speculative, although suchneurons have been observed in motorneurons in anesthetized rats13. In any case,Koulakov et al. use two models for bista-bility: one in which the voltage-dependentinward current is supplied by NMDAchannels, the other in which there is anactivity-dependent drive that is suppliedby small groups of garden-variety neurons.

Whether or not their model applies toreal neural integrators, however, is still farfrom decided. From a theoretical point ofview, the model must be made more real-

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