AR245-NE28-14 ARI 21 March 2005 18:15 V I E E W R S First published online as a Review in Advance on March 22, 2005 I E N C N A D V A Neural Network Dynamics TimP.Vogels, Kanaka Rajan, and L.F. Abbott Volen Center for Complex Systems and Department of Biology, Brandeis University, Waltham, Massachusetts 02454-9110; email: [email protected] Annu. Rev. Neurosci. Key Words 2005. 28:357–76 balance, memory, signal propagation, states, sustained activity doi: 10.1146/ annurev.neuro.28.061604.135637 Abstract Copyright c 2005 by Neural network modeling is often concerned with stimulus-driven by "Melville Library, SUNY Stony Brook" on 03/22/05. For personal use only. Annual Reviews. All rights reserved responses, but most of the activity in the brain is internally generated. 0147-006X/05/0721- Here, we review network models of internally generated activity, fo- 0357$20.00 cusing on three types of network dynamics: (a) sustained responses to transient stimuli, which provide a model of working memory; (b) os- Annu. Rev. Neurosci. 0.0:${article.fPage}-${article.lPage}. Downloaded from arjournals.annualreviews.org cillatory network activity; and (c) chaotic activity, which models com- plex patterns of background spiking in cortical and other circuits. We also review propagation of stimulus-driven activity through sponta- neously active networks. Exploring these aspects of neural network dynamics is critical for understanding how neural circuits produce cognitive function. www.annualreviews.org · Network Dynamics 357 AR245-NE28-14 ARI 21 March 2005 18:15 Understanding how neural circuitry gen- Contents erates complex patterns of activity is challeng- ing, and it is even more difficult to build mod- INTRODUCTION ..................358 els of this type that remain sensitive to sensory FIRING-RATE AND SPIKING input. In mathematical terms, we need to un- NETWORK MODELS ...........359 derstand how a system can reconcile a rich Firing-Rate Networks ..............360 internal state structure with a high degree of Integrate-and-Fire Networks .......360 sensitivity to external variables. This problem FORMS OF NETWORK is far from solved, but here we review progress DYNAMICS ......................362 that has been made in recent years. Rather Sustained Activity ..................364 than surveying a large number of models and Oscillations ........................365 applications, we illustrate the existing issues Chaotic Spiking Networks..........366 and the progress made using two basic models: SIGNAL PROPAGATION ...........368 a network model described in terms of neu- Avalanche Model ..................368 ronal firing rates that exhibits sustained and Synfire Chains .....................369 oscillatory activity and a network of spiking Propagation of Firing Rates ........372 model neurons that displays chaotic activity. DISCUSSION .......................372 We begin the review with a discussion of sustained responses to transient stimuli. Neu- ronal activity evoked by a transient stimulus often continues beyond the period of stim- INTRODUCTION ulus presentation and, in cases where short- We generate most of our thoughts and behav- term memory of the stimulus is required for iors internally, but our actions can be modified a task, such sustained activity can last for tens drastically by small changes in our perception of seconds (Wang & Goldman-Rakic 2004). of the external environment. Stated another Neuronal firing at a constant rate is a form of way, the neural circuits of the brain perpetu- internally generated activity known as fixed- ally generate complex patterns of activity with point behavior. This time-independent activ- an extraordinarily rich spatial and temporal ity is too simple to address the issue of how structure, yet they remain highly sensitive complex patterns of activity are generated. to sensory input. The majority of modeling On the other hand, these models provide an in neuroscience is concerned with activity excellent example of the problem of making that is driven by a stimulus. Such models internally generated activity sensitive to exter- are constructed to account for sensitivity, nal input because, to be of any use in a mem- selectivity, and other features of neuronal ory task, self-sustained neural activity must be responses to sensory input (reviewed, for sensitive to those aspects of the stimulus that by "Melville Library, SUNY Stony Brook" on 03/22/05. For personal use only. example, in Dayan & Abbott 2001). In the are being remembered (Compte et al. 2000, absence of that input, neurons in these models Seung et al. 2000). are typically silent. Although this approach From sustained activity, we move on to Annu. Rev. Neurosci. 0.0:${article.fPage}-${article.lPage}. Downloaded from arjournals.annualreviews.org has had considerable success in accounting a discussion of oscillations. Oscillations are for response properties in primary sensory a widespread feature of neural systems, and areas, such as the early visual system, it clearly oscillating neural network models have been cannot account for the majority of activity studied extensively (Marder & Calabrese in the brain, which is internally generated. 1996, Buzsaki & Draguhn 2004). We illustrate This review is devoted to modeling work at this form of network activity by modifying a the other extreme: models that produce their model with self-sustained activity. own activity, even in the absence of external Our next topic covers large networks of input. spiking model neurons that display complex 358 Vogels · Rajan · Abbott AR245-NE28-14 ARI 21 March 2005 18:15 chaotic activity. In these networks, excitation networks is difficult to achieve because of two and inhibition are balanced so that they nearly sources of instability. First, signals tend to cancel, and neuronal firing is driven by fluc- either grow or shrink in amplitude as they tuations that transiently spoil this cancella- propagate from one group of neurons to an- tion (Shadlen & Newsome 1994, Tsodyks & other. Rather precise tuning is required to Sejnowski 1995, Troyer& Miller 1997). Neu- prevent signals from either blowing up or ronal responses recorded in vivo are highly decaying away before they reach their tar- variable (Burns & Webb 1976, Dean 1981, gets. This problem is well illustrated in a Softky & Koch 1993, Holt et al. 1996, simple avalanche model of propagation that Anderson et al. 2000), and it has long been we review (Harris 1963, Zapperi et al. 1995, recognized that some form of “noise” has to de Carvalho & Prado 2000, Beggs & Plenz be included if models of such responses are to 2003). Second, signal propagation in neural match the data (see, for example, Usher et al. networks can lead to unrealistic large-scale 1994). In models, this noise is often added synchronization of neuronal firing (Marsalek from a random external source, such as a ran- et al. 1997, Golomb 1998, Burkitt & Clark dom number generator, which does not match 1999, van Rossum et al. 2002, Reyes 2003, what happens in real neural circuits. Although Litvak et al. 2003). Avoiding this problem re- neurons are subject to thermal fluctuations quires the introduction of noise, which leads that act like an external source of noise, it ap- us back to sparsely coupled networks of spik- pears that most of the variability in cortical ing neurons that can generate the required circuits comes from activity generated within noise internally. As discussed below, signal the neural circuits themselves (Arieli et al. propagation and sensitivity to input remain 1996). Sparsely connected networks of spik- significant challenges to our understanding of ing model neurons can generate what looks neural network dynamics. like random, noisy activity without the need for any external source of randomness (van Vreeswijk & Sompolinsky 1996, 1998; Amit FIRING-RATE AND SPIKING & Brunel 1997; Brunel 2000; Mehring et al. NETWORK MODELS 2003; Ahmadi et al. 2004). This is a significant The power of present-day computers permits achievement toward the goal of understand- simulation of large networks, even in cases ing the dynamics of complex neuronal activity. when the individual neurons are modeled in Spiking network models go a long way to- considerable detail. Of course, there is a trade- ward solving the problem of producing com- off between the amount of detail that can be plex, self-sustainedpatterns of activity, but devoted to modeling each individual neuron they fail at accounting for the input sensitiv- (or each synapse, which is even more costly ity of biological networks. Returning to this owing to their larger numbers) and the size by "Melville Library, SUNY Stony Brook" on 03/22/05. For personal use only. problem, we conclude this review by exam- and complexity of the network that can be ining studies of signal propagation in neural simulated. A common compromise is to use a networks. For a network to be sensitive to relatively simple spiking model, the integrate- Annu. Rev. Neurosci. 0.0:${article.fPage}-${article.lPage}. Downloaded from arjournals.annualreviews.org external input, the activity generated by that and-fire model, to describe each neuron. This input must propagate through the network. allows simulations of networks with tens or There has been considerable discussion about even hundreds of thousands of neurons. the different ways that information might be Such networks are complex dynamical sys- encoded by neural activity. Tobe viable, a cod- tems involving the numerical integration of ing scheme must support propagation of in- many thousands of coupled differential equa- formation from one brain region to another tions. In computer simulations, as opposed to (Diesmann et al. 1999, van Rossum et al. experiments, any variable in any neuron or 2002). Propagation of signals across neural synapse of the network can be monitored and www.annualreviews.org · Network Dynamics 359 AR245-NE28-14 ARI 21 March 2005 18:15 manipulated. Nevertheless, characterizing tal synaptic input is obtained by summing the and understanding what is going on at the net- contributions of all the presynaptic neurons work level can be difficult. Furthermore, time (the sum over j in Equation 1).