Appendix F: Theoretical Approaches to Neuroscience: Examples from Single Neurons to Networks

• Single-Neuron Models Allow Study of the Integration of Synaptic Inputs and Intrinsic Conductances

  • Neurons Show Sharp Threshold Sensitivity to the Number and Synchrony of Synaptic Inputs in Quiet Conditions Resembling In Vitro

  • Neurons Show Graded Sensitivity to the Number and Synchrony of Synaptic Inputs in Noisy Conditions Resembling In Vivo

  • Neuronal Messages Depend on Intrinsic Activity and Extrinsic Signals

• Network Models Provide Insight into the Collective Dynamics of Neurons

  • Balanced Networks of Active Neurons Can Generate the Ongoing Noisy Activity Seen In Vivo

  • Feed-forward and Recurrent Networks Can Amplify or Integrate Inputs with Distinct Dynamics

  • Balanced Recurrent Networks Can Behave Like Feed-forward Networks

  • Paradoxical Effects in Balanced Recurrent Networks May Underlie Surround Suppression in the Visual Cortex

  • Recurrent Networks Can Model Decision-Making

In one way or another all scientific researchers construct models. The role of theory is to develop, refine, and investigate these models to uncover new insights and to make new predictions. Theoretical (or computational) neuroscience explores the application of mathematical and computational methods to the study of neural systems.

Defining models of neural systems in mathematical terms assures that the models are self-consistent, with clearly revealed assumptions and limitations. Formulating models mathematically also allows the powerful techniques of mathematics and physics and the extraordinary capacity of modern computers to be used to understand model behavior. This power allows the implications of the models to be worked out fully, revealing features that are surprising and nonintuitive.

What makes a model good? Clearly it must be based on biological reality, but modeling necessarily involves an abstraction of that reality. It is important to appreciate that a more detailed model is not necessarily a better model. A simple model that allows us to think about a phenomenon more clearly is more powerful than a model with underlying assumptions and mechanisms that are obscured by complexity. The purpose of modeling is to illuminate, and the ultimate test of a model is not simply that it makes predictions that can be tested experimentally, but whether it leads to better understanding. No matter how detailed, no model can capture all aspects of the phenomenon being studied. As theoretical neuroscientist Idan Segev has said, borrowing from Picasso's description of art, modeling is the lie that reveals the truth.

Appendix E introduced two ideas that are central in theoretical neuroscience: the perceptron and the attractor. In this appendix we extend the discussion of theoretical neuroscience by considering the effect of synaptic input on neuron models that produce action potentials, by showing how inhibition and excitation can interact in network models to produce interesting and nonintuitive effects, and by presenting a model of decision-making. These examples convey a sense of how theoretical neuroscience research is done, and how it can shed light on the types of computations that neurons and networks of neurons can perform.

Models of single neurons include some of the most detailed, accurate, and complex simulations performed in the field of theoretical neuroscience. Single-neuron modeling is used to reveal how the many different voltage- and ligand-dependent conductances expressed by a neuron interact to generate patterns of activity ranging from transient and sustained firing to the production of subthreshold oscillations and action-potential bursts. Because membrane conductances are nonlinear, interactions between them can be complex. Modeling studies allow these interactions to be mapped out and understood.

Another major focus of single-neuron modeling is to understand the functional consequences of the complex and beautiful morphology of neurons. One goal of this work is to understand how the specific location of a synapse or group of synapses on the dendritic arbor, in interaction with the voltage-dependent conductances in dendrites, affects the efficacy of synapses to drive a neuron to fire. Synaptic plasticity is also a major target of dynamically detailed modeling of neurons. In this regard backpropagating action potentials, which provide information to the synapses about the timing of somatic action potentials, are of particular interest. This work, and of course its experimental counterpart, has revealed the roles of nonlinear summation, subthreshold boosting, and dendritic spiking in the integration of inputs across the dendrite.

Single-neuron modeling studies are also valuable for understanding the effects of complex patterns of synaptic inputs on neuronal activity. In an experimental setting it is difficult to control large numbers of synaptic inputs precisely enough to perform analogous studies. Neurons are often studied in fairly inactive tissue slices or isolated in cell cultures. In vitro experiments can tell us how a particular neuron responds to the activation of one or a few synapses (from paired recordings, for example) or to synchronous activation of many synapses (through extracellular stimulation). Studying neurons with controlled synaptic input is ideal for exploring their basic electrophysiological characteristics, but not for understanding how they operate in vivo. We typically want to know how the neuron reacts to the complex input patterns that it receives from thousands of afferent fibers in a live animal. Theoretical studies allow us to bridge the gap between what we know from reduced experimental protocols and what we need to know to understand functioning neural circuits: the response properties of neurons in their natural environment.

Appendix E considered a neuron model in which synchronous presynaptic action potentials are counted and compared with a threshold to determine output. Here we examine how the large quantity of constant synaptic input in vivo modifies this picture, focusing on the effects of synchronization and timing of presynaptic inputs on the response of a neuron. Action potentials are also known as spikes, and we use the two terms interchangeably.

The role of spike synchronization in neural coding and signaling is actively debated. Synchronization of spikes has been proposed as a mechanism by which attention might enhance signals of particular interest, based on the observation that synchronous action potentials drive a neuron more strongly than the same number of action potentials spread over time. However, it is unlikely that all inputs to a neuron would ever be synchronized. Instead, we assume that the neuron receives a "background" of many asynchronous inputs, and that a small number of these inputs, which we call the signaling inputs, become synchronized by a particular signal or drive. Thus we need to study the effects of various degrees of synchronization in subsets of signaling inputs in a background of asynchronous input. Studying the effects of spike timing in more realistic conditions allows us to assess more accurately the role that timing may play in neural coding of information and in synaptic plasticity.

We will compare the sensitivity of neurons to the numbers and timing precision of synchronous signaling inputs in a quiet slice-like setting and in an active environment of asynchronous background inputs resembling that found in vivo. We will use a simple but useful model of the postsynaptic neuron known as the integrate-and-fire model in which inputs are integrated until a threshold voltage is reached, at which point a spike is emitted and the voltage is then reset to a lower value (Box F–1). By studying the responses of this model neuron we can compare the impacts of spike synchronization with and without asynchronous background input.

Neurons Show Sharp Threshold Sensitivity to the Number and Synchrony of Synaptic Inputs in Quiet Conditions Resembling In Vitro

We first explore the sensitivity of our integrate-and-fire model neuron to the number of synchronous signaling inputs it receives in the absence of any background asynchronous input. Under these conditions 43 synchronous presynaptic inputs are required to elicit an action potential (Figure F–1A).

The number 43 depends on the particular model parameters we have chosen, but a more general property is also revealed. For the model in Appendix E the probability of producing a spike rises as a sharp step from 0 to 1, meaning that fewer than 43 synchronous presynaptic action potentials will never produce a response, whereas 43 or more will always produce a postsynaptic spike. Thus, when synchronous input appears alone without any asynchronous inputs, the neuron has a sharp threshold sensitivity to the number of synchronous action potentials it receives.

How sensitive is the neuron to timing of the signaling synaptic inputs? To address this question we look at what happens when the synchronous action potentials are spread over a finite period of time, thereby slightly desynchronizing them. When 50 input spikes are spread evenly over intervals ranging from zero to 50 ms, the model neuron responds only if the 50 input spikes are contained within an interval of 20 ms or less (Figure F–1B), indicating that it is quite sensitive to the interval over which the 50 input spikes are spread. Not surprisingly, the 20 ms interval matches the membrane time constant of the model neuron. Thus in the absence of asynchronous input this model neuron is highly sensitive to both the number and timing of its synchronous or nearly synchronous inputs.

Neurons Show Graded Sensitivity to the Number and Synchrony of Synaptic Inputs in Noisy Conditions Resembling In Vivo

What is the effect of background synaptic activity on the sensitivity of a neuron to synchronous signaling input? To address this question we again introduce either synchronous or slightly desynchronized signaling input, but this time we add it to ongoing asynchronous background input.

The nature and impact of ongoing background activity, which varies across brain areas, is a subject of intense experimental and theoretical investigation. Experiments that use different recording techniques do not always agree on the spontaneous firing rates even of neurons in the same area, and conflicting values are reported for the total synaptic conductance produced by spontaneous activity. Nevertheless, in studies of the cerebral cortex all researchers agree that background activity is large enough to cause significant fluctuations in the membrane potentials of recorded neurons, sometimes large enough to drive firing.

The background synaptic input we are discussing contains both excitatory and inhibitory components. In some cortical areas the background input appears as occasional bursts amidst relative quiet. In many other cortical areas the background input drives constant, ongoing voltage fluctuations. In these areas there is evidence that the excitatory and inhibitory components are individually quite large but roughly balance or cancel each other out. This cancellation is subject to fluctuations in which excitatory or inhibitory drive momentarily dominates. In the next section we illustrate how a network of deterministic model neurons can generate such fluctuations spontaneously. Here we obtain similar fluctuations by randomly generating presynaptic action potentials.

Ongoing background activity affects how a neuron integrates its signaling inputs. It also acts as a source of noise that usually limits, but in some cases enhances, the sensitivity and precision of responses to signaling inputs. For the single-neuron model we are using the asynchronous background synaptic input has three primary impacts. First, it alters the neuronal membrane potential, in our case raising it from a resting value of 20 mV below threshold to approximately 5 mV below threshold. This effect is caused by the fact that the cancellation or balance between excitation and inhibition is, even on average, only approximate. Because the model neuron is now closer to threshold, it becomes more sensitive to additional input.

Second, the continual activation of both excitatory and inhibitory synaptic conductances by ongoing input increases the overall conductance of the neuron. This lowers its effective time constant, allowing for quicker responses, and also reduces the sensitivity of the neuron to other synaptic input. However, for our model parameters the lowering of sensitivity is more than compensated by the fact that the baseline membrane potential is raised by 15 mV.

Third, background synaptic input introduces fluctuations in the membrane potential that generate a low rate of background firing and add a level of noise to responses to signaling inputs (Figure F–1C).

To study the effect of synchronous signaling input in the presence of this background, we add a number of extra presynaptic spikes at a particular point in time. The addition of 20 synchronous spikes produces a postsynaptic response on 4 out of 10 trials, whereas adding 40 synchronous inputs generates a spiking response on 7 of 10 trials (Figure F–1A). By accumulating results like these over many repeated trials, we can determine how often (on what fraction of trials or, equivalently, with what probability) a postsynaptic action potential is evoked within a short time interval (10 ms) following the synchronous presynaptic input.

Recall that in the absence of background input the model neuron showed sharp threshold sensitivity to the number of synchronized input spikes. This sharp threshold is lost when background input is added, but it is replaced by a higher sensitivity to small numbers of synchronized presynaptic action potentials (Figure F–1A). Rather than requiring at least 43 synchronous excitatory inputs to generate an output, the probability that the neuron will fire in the presence of background activity rises smoothly from zero. The neuron has a significant probability of responding to as few as 10 synchronous presynaptic action potentials.

Ongoing asynchronous background input also alters the probability of evoking a postsynaptic action potential when presynaptic action potentials are not perfectly synchronous but are spread uniformly over a finite stimulation time. Just as the neuron lost its step-like threshold for the number of presynaptic action potentials required to elicit a spike, it also loses its sharp sensitivity to the relative timing of these inputs (Figure F–1B). The neuron responds approximately 30% of the time to 50 additional action potentials even when they are spread over 50 ms. Thus the results in Figures F–1A and F–1B show that background input makes the neuron more sensitive to small numbers of presynaptic spikes but less sensitive to their synchrony (the degree to which they are tightly grouped in time).

Neuronal Messages Depend on Intrinsic Activity and Extrinsic Signals

The issue of whether to think of spike timing or firing rates as the dominant carrier of the "neural code" has been debated extensively, but it is often difficult to understand exactly what is being debated. Action potentials in a particular pathway carry information by virtue of their times of occurrence; they have no other attribute that can be varied systematically.

A steady signal can be adequately represented by neuronal firing rates because there is no temporal structure for spike times to encode. For rapidly fluctuating signals, however, spikes times can be important because they carry information about the timing and amplitude of such fluctuations. Either way we must remember that neurons respond in the context of the active environment in which they operate.

As we see from Figure F–1, inferences made from observations of neurons in a silent environment (without background synaptic input) may underestimate the responsiveness of a neuron to weak excitation. Similarly the ability of the neuron to sharply delineate different levels of input may be overestimated and the neuron's sensitivity to spike timing exaggerated.

Models of networks of neurons, or neural circuits, help us understand how large populations of neurons can work together to perform tasks. Using models, theorists have learned how networks can sustain particular patterns of reverberating activity or generate oscillatory or chaotic firing.

Network models address such issues as how neurons in primary sensory areas produce their responses, how cortical maps form and are modified by sensory experience, how populations of neurons represent information collectively and combine multisensory data, how objects are identified and classified, how memories stored in modified synaptic strengths can be read out, how motor responses are generated, and how evidence is integrated and used to make decisions.

Appendix E discusses basic ideas of how feed- forward circuitry (circuitry with no synaptic loops) may underlie the response properties of simple and complex cells in the primary visual cortex, and how recurrent circuitry (circuitry with synaptic loops) can create a cell assembly with strong mutual excitation among its neurons, allowing the assembly's activity to persist in the absence of a stimulus (thus forming an attractor). Here we continue these discussions with examples from network modeling.

We will examine how recurrent loops change both the gain and dynamics of responses to inputs, how a fully recurrent network can have hidden within it an effectively feed-forward structure that allows changes in gain to be separated from changes in dynamics, and how these ideas can give insight into aspects of visual cortex function. We also show how the basic idea of attractors can be extended to create a model of decision-making. Before we consider these examples, we show what can happen when model neurons like the one discussed previously are linked together into a network.

Balanced Networks of Active Neurons Can Generate the Ongoing Noisy Activity Seen In Vivo

When constructing and analyzing large networks it is typically not practical to model the constituent neurons with a high level of detail. Instead, the multitude of synaptic conductances and the complexity of dendritic morphology seen in real neurons are usually distilled down to a bare minimum.

In network models individual neurons are often modeled as integrate-and-fire neurons (see Box F–1) but may also be modeled as firing-rate neurons, meaning that only the rate at which a neuron fires is modeled, and not the timing of individual spikes (Box F–2). We employ both neuron models here. Simplifying the descriptions of individual neurons allows us to focus on effects that arise through network interactions.

What happens if we link together a large number of integrate-and-fire neurons through excitatory and inhibitory synapses? Such models typically involve thousands or even hundreds of thousands of neurons connected randomly and sparsely so that the probability of any two neurons being connected is less than 10%. Excitatory and inhibitory neurons are included in roughly the 4:1 proportion seen in cortical circuits.

The activity in such networks takes a number of different forms as shown by Carl van Vreeswijk and Haim Sompolinsky and by Nicolas Brunel. When the synaptic connections are weak, the bulk of the neurons are silent; but some fire in steady, regular sequences of action potentials that are not synchronized with the activity of other neurons of the network (Figure F–2A). As the strengths of the synapses, both excitatory and inhibitory, are increased, the network can transition into a state where the silent neurons start to fire, and the action potentials appear in irregular, asynchronous patterns (Figure F–2B). This form of activity provides a model of the background activity seen in real neural circuits.

The irregular, asynchronous activity depends on the sparseness of the synaptic connections in the network and on a balance between excitatory and inhibitory inputs received by a cell. These inputs arise from many other excitatory and inhibitory neurons that are themselves firing irregularly. Overall these inputs are balanced—on average, excitation and inhibition cancel—but constant fluctuations in input drive the cell to fire at irregular times. Thus the network self-consistently maintains irregular firing in all of its neurons. If the balance of excitation and inhibition is not maintained, a network-model form of epilepsy can arise (Figure F–2C). In this case the asynchronous activity is interrupted by gaps and periods of synchronous firing across the population. It is interesting that network models of ongoing spontaneous activity suffer quite easily from seizure-like activity, just like real neural circuits.

Irregular, asynchronous firing, much like the background firing seen in many cortical areas, can arise in network models with different patterns of synaptic connectivity, such as when the probability of connection between two neurons decreases with the distance between them or with the difference between their selectivities to stimulus properties. The major requirement is that inhibition must balance excitation, so that the mean input cannot drive the cell to fire and instead firing is induced by input fluctuations. In addition, connectivity should be sparse so that the firing of different cells does not become synchronized.

What can structured network circuitry achieve? In the following discussion we provide a few illustrative answers to this question. For this analysis we switch from a network of spiking model neurons to one in which the activities of network neurons are described by firing rates (Box F–2). Networks of spiking model neurons can display patterns of activity that cannot be reproduced by firing-rate models, but there is good agreement between the two types of models for the steady-state asynchronous activity we discuss here and in the following sections. Describing neuronal responses in terms of firing rates makes the mathematical analysis of neural networks much easier.

Feed-forward and Recurrent Networks Can Amplify or Integrate Inputs with Distinct Dynamics

The relative roles played by feed-forward and recurrent circuitry in shaping neuronal responses is a subject of debate in neuroscience. For example, neurons in layer IV of the primary visual cortex receive afferent inputs from the lateral geniculate nucleus of the thalamus, which relays visual information from the eyes, but they also receive abundant innervation from other cortical neurons. Are the tuning properties of these neurons—the dependence of their firing rates on stimulus parameters such as the orientation of a light or dark edge—determined mainly by feed-forward input from the lateral geniculate nucleus, or are they strongly shaped by recurrent cortical feedback?

The answer is not immediately obvious because feed-forward circuits and circuits in which tuning is strongly shaped by recurrence (recurrent networks) can produce the same types of response selectivity or tuning. Intracellular recording in vivo, in which voltage responses and their changes under experimental perturbations are studied, can more clearly distinguish the two types of circuits. However, given only firing rate responses, the differences between feed-forward and recurrent circuits are easiest to detect by examining response dynamics rather than static response properties. For this reason, here we discuss the dynamics of network responses in various forms of feed-forward and recurrent circuits.

A feed-forward circuit can modify input signals, creating a wide variety of response selectivities or amplifying a weak input without significantly altering response dynamics. For example, if one set of neurons forms strong excitatory synapses with another set, a weak input to the first set can yield a strong response in the second. This occurs with only a small dynamic change, namely the small delay required for the first set of neurons to integrate the input and produce spikes that propagate to drive the second set of neurons.

Similarly, recurrent circuits can modify input signals, but typically with larger dynamic changes. An example of this is a circuit that amplifies its inputs through recurrent excitatory loops. Consider a population of neurons that excites itself. The population's response to an external input can be significantly amplified because the recurrent excitation adds to the external drive. However, unlike feed forward amplification the recurrent excitation is accompanied by a general slowing of the population's response dynamics, which arises as follows.

A population responds to a pulse of input with a pulse of activity that then decays away. The recurrent excitation adds back some of the activity that would otherwise decay, slowing the decay of the population's activity. The population's response to a sustained input is similarly slowed; the ultimate response to input is amplified by a factor roughly equal to the degree of slowing (ie, a threefold slowing yields a threefold amplification) (Figure F–3A). This can be understood by thinking of the sustained input as a continuous sequence of pulse inputs. We imagine that the response to the sequence of pulses is just the sum of the responses to each individual pulse. (This is not quantitatively correct but provides a useful qualitative representation.) Because individual pulse responses decay more slowly, the overall level to which they sum is increased, but the rise of activity to this level occurs more slowly.

A network with recurrent excitation and inhibition can also have inhibitory loops through which a population of neurons inhibits itself. In this case responses are sped up rather than slowed down, and they are reduced in amplitude (Figure F–3B). The reduction in response amplitude occurs because the decay of the response to a pulse of input is accelerated: In addition to the decay that would otherwise occur, inhibition subtracts even more from the activity. Because the responses to individual pulses decay more quickly, the overall level to which they sum is decreased, but the rise of activity to this level occurs more quickly.

If recurrent excitation is increased to the point where activity set up by a transient input can sustain itself indefinitely, decay does not occur at all and the response is infinitely slowed. This requires fine-tuning of network parameters. The resulting circuit, known as an integrator network, has some interesting properties. The response of an integrator network to a transient pulse of input is a change in firing rate that lasts forever in the absence of further input but which becomes part of an ongoing integral if further input is applied (Figure F–3C). If the transient excitation in the network is not perfectly tuned but instead is slightly weaker, the input produces a change in firing rate that decays very slowly. Such approximate integrators are used to model neural circuits that remember signals.

What happens if the amount of excitation is increased beyond the point of perfect tuning that achieves an integrator? The examples shown in Figure F–3 all start and end in a resting state with zero activity. When excitation is overly strong, such a resting state, whether or not it is characterized by zero activity, becomes unstable. This means that after any small perturbation, induced for example by a transient input, the system will drive itself further away from this state rather than relaxing back to it. Nonlinear processes ultimately stabilize a new pattern of activity, which is determined primarily by the network's own recurrent excitation rather than by the input and which can be self-sustaining in the absence of an input.

Fixed patterns of activity established and maintained by recurrent circuitry are attractors, as described in Appendix E. Attractors are often used as models for the persistent neural activity thought to hold items in working memory. If many different activity patterns are each strongly self-excitatory, a network can generate far more complex, chaotic dynamics. It has been argued that this can create a rich set of temporal patterns in areas of cortex such as primary motor cortex that can be harnessed by signals from motor planning areas to drive complex movement patterns.

Balanced Recurrent Networks Can Behave Like Feedforward Networks

Anatomical and electrophysiological studies of neurons in layer IV of the primary visual cortex (and in other sensory areas) have revealed many more recurrent than feed-forward connections in this layer. This discovery might seem to rule out feed-forward networks as relevant models of cortical circuits. However, function need not follow numbers.

A prominent hypothesis, supported by considerable evidence, is that the feed-forward inputs provide the driving input to layer IV neurons while the recurrent inputs amplify and modulate but do not drive responses. Feed-forward circuits are also relevant in another way: Under appropriate circumstances, strongly recurrent circuits can act effectively in a feedforward manner.

To see how this works, we study a network consisting of two coupled populations, one excitatory and one inhibitory. To simplify the discussion we let the two projections from the excitatory population (to itself and to the inhibitory population) have the same strength, and likewise the two inhibitory projections. Thus we can speak of the strength of excitation (or inhibition)—meaning excitation (or inhibition) onto both excitatory and inhibitory neurons—without having to distinguish multiple projections of each type. Although this simplification dictates the specific results, it does not affect the overall conclusions as to how recurrent circuits can act in a feedforward manner and thus dissociate amplification from changes in dynamics.

We suppose the network is at a fixed point, meaning that there are steady excitatory and inhibitory firing rates in response to a steady feed-forward input, and that this fixed point is stable—after small transient perturbations the network will return to the fixed point. We suddenly increase the level of feed-forward input to the excitatory population (Figure F–3D). In this recurrent network the excitatory activity is amplified by the recurrent circuitry. Surprisingly, however, the time course of the increase is little different from that without recurrent circuitry. With different parameters the recurrently amplified response can exactly match the timing of, or even be faster than, the response without recurrent circuitry. Thus the recurrent circuit can amplify responses without any slowing of the temporal dynamics.

Apparently the mechanism of amplification is different from the recurrent amplification we considered previously. What is this mechanism? At the fixed point the difference between the excitation and inhibition received by a population is exactly that required to sustain the population's firing rates. Any change in the balance between excitation and inhibition drives a change in the firing rates until a new balance is restored. Increasing the feed forward input to the excitatory population shifts the balance toward excitation, thus driving up both excitatory and inhibitory firing rates. Similarly, an excess of inhibition drives down both excitatory and inhibitory firing rates.

Let us represent the firing rates as differences from the fixed-point firing rates. In this representation firing rates may be either positive or negative. We can then formally represent any pattern of excitatory and inhibitory firing rates as a weighted combination of two activity patterns. In the differential pattern excitatory and inhibitory cells have equal and opposite firing rates. In the common pattern excitatory and inhibitory cells have identical firing rates. Given some excitatory and inhibitory firing rates, if we weight the common pattern so that its common firing rate is the average of the excitatory and inhibitory rates, and weight the differential pattern so that it captures the difference between excitatory and inhibitory firing rates, then the sum of the two weighted patterns equals the given firing rates.

The advantage of expressing the activities in terms of these two patterns is that it allows a deeper insight into the dynamics. A shift in the network's balance toward excitation involves an increase in the size of the differential pattern, that is, the signed difference between excitatory and inhibitory firing rates increases. This imbalance in turn drives an increase in the size of the common pattern, that is, both excitatory and inhibitory firing rates increase. Similarly, a shift in the balance toward inhibition involves a decrease in the differential pattern, which decreases the common pattern. The network thus behaves precisely as it would if the differential activity pattern made a feedforward synaptic connection to the common activity pattern, that is, imbalances drive balanced responses. Furthermore, there is no corresponding feedback from the common pattern onto the differential pattern. Balanced responses do not drive imbalances, nor does the differential pattern act on itself—imbalances do not drive further imbalance.

Thus the network of Figure F–3D, which appears to be fully recurrent when viewed in terms of the neurons, can be seen to have a hidden feed-forward structure when viewed in terms of differential and common activity patterns (it is hidden in the sense that it is not readily apparent in the synaptic connectivity of the network). This feed-forward pathway allows one activity pattern to excite the other without feedback (Figure F–4). The amplification driven by this "hidden" feedforward connection occurs with little dynamical slowing, whereas amplification as a result of a self-excitatory loop is achieved at the cost of slowing of the dynamics. This description is mathematically precise when the relationship of a neuron's firing rate to its input (see Box F–2) can be taken to be linear, and it provides a useful intuition for understanding this type of network behavior more generally.

The strength of the amplification depends on two factors. One is the strength of the feed-forward connection from the differential to the common pattern, which is given by the sum of the excitatory and inhibitory synaptic strengths. The other is the strength of any remaining feedback loops. If we assume that inhibition is stronger than excitation, this results in a loop by which the common pattern inhibits itself (Figure F–4), because raising both excitatory and inhibitory rates produces a net inhibition of both the excitatory and inhibitory populations. This self-inhibition of the common pattern suppresses rather than amplifies responses; its strength is given by the difference between excitatory and inhibitory synaptic strengths. Thus the strongest amplification arises when both excitation and inhibition are strong but reasonably balanced.

This form of amplification can occur for each of many different spatial patterns of activity in a network of many neurons. In each spatial pattern an imbalance of excitatory and inhibitory activity can provide feed-forward drive to common excitatory and inhibitory activity, with different spatial patterns having different strengths of this drive and thus different degrees of amplification. This mechanism has been proposed to underlie observations of spontaneous activity in the primary visual cortex of anesthetized cats (that is, activity in the absence of a visual stimulus).

Despite the absence of a visual stimulus, spatial patterns of activity resembling responses to structured visual inputs make a larger contribution to the overall spontaneous activity than patterns unrelated to these visual responses. If we think of spontaneous activity as being driven by unstructured inputs to the visual cortex that equally drive many different patterns, then the patterns that resemble visually driven responses are being amplified by the visual cortical circuit more than other patterns. What is the mechanism underlying this amplification?

Preliminary analysis suggests that the dynamics of the amplified patterns are not significantly slowed. In a model network with balanced excitatory and inhibitory connections that preferentially target cells with similar orientation selectivity, the spatial patterns that resemble visually driven responses have the largest effective feed-forward weights and so are amplified relative to other spatial patterns, without dynamical slowing, by the mechanism shown in Figure F–3D. This provides a possible explanation for the amplification of activity patterns that resemble visual responses in the absence of a visual stimulus.

Paradoxical Effects in Balanced Recurrent Networks May Underlie Surround Suppression in the Visual Cortex

In this section we discuss another effect that can arise in networks in which excitation and inhibition are both strong but relatively balanced. We consider a network that satisfies two criteria.

First, the excitatory recurrence is strong enough to make the excitatory network unstable by itself. That is, if the network is at a stable fixed point, and if inhibitory firing rates were kept frozen at their fixed-point levels, then after small perturbations of excitatory firing rates the excitatory network would drive itself even further away from its fixed-point rates. Second, feedback inhibition (inhibition driven by the excitatory cells) stabilizes the network. A slight change in excitatory firing rates drives a sufficient change in inhibitory firing rates to push the excitatory rates back to their fixed-point levels, despite the tendency of the excitatory network to "run away" on its own.

We refer to a network meeting these two criteria as an inhibition-stabilized network. The strong recurrent excitation received by excitatory cortical neurons and the instability of cortical activity when inhibition is blocked suggest that cortical circuits may indeed be stabilized in this way.

Inhibition-stabilized networks provide a possible explanation for a paradoxical experimental observation. The region of visual space within which an appropriate visual stimulus can elicit a response in a neuron in the primary visual cortex (V1) is known as the center region of the cell's receptive field. For many V1 neurons, increasing the size of the stimulus so that it also covers the surrounding region (the surround) reduces the response, a phenomenon known as surround suppression. However, a stimulus covering only the surround and not the center yields no response.

It is believed that stimulation of the center of a neuron's receptive field drives external excitatory input relayed from the eyes to both excitatory and inhibitory neural populations in the local cortical circuit, whereas a surround stimulus excites neighboring regions of cortex that send excitation more strongly to the inhibitory population within the local circuit. We might therefore expect that a stimulus in the surround should increase firing in the local inhibitory population, which in turn would suppress responses in the excitatory population.

Instead, experiments by David Ferster and colleagues indicate that a surround stimulus reduces both the excitation and the inhibition that a V1 neuron receives. That is, surround suppression is actually mediated by a reduction in the excitation of a cell, which has a larger effect than a concurrent reduction in inhibition. The reduction in inhibition suggests that the firing rate of the inhibitory population, like that of the excitatory population, is reduced by the surround stimulus, and this has been directly confirmed by Xue-Mei Song and Chao-Yi Li. Thus we arrive at the paradoxical result: A surround stimulus that is believed to drive external excitation to an inhibitory population causes a net decrease in the inhibitory population's firing rate.

This paradox can be explained by the presence in the cortex of strong recurrent excitation that is stabilized by inhibition, as shown in a model constructed by Misha Tsodyks and colleagues in a different context. The inhibitory neurons receive such strong drive from the excitatory neurons that their activity is determined more by the excitatory neurons than by any external input.

To see this, consider an inhibition-stabilized network composed of two populations of neurons, one excitatory and the other inhibitory (as in Figure F–3D), each initially firing at steady rates in response to a constant center stimulus (Figure F–5A). Adding a surround stimulus provides additional excitation to the inhibitory neurons from external sources. This transiently increases inhibitory neurons firing (Figure F–5B), which in turn lowers the firing rates of the excitatory neurons in the network (Figure F–5C), causing a withdrawal of recurrent excitatory input to the inhibitory neurons. Precisely when the network is inhibition-stabilized, this withdrawal of recurrent excitatory input to the inhibitory neurons exceeds the increase in excitation from external sources that started the process, so that the ultimate result, paradoxically, is that inhibitory firing rates are also lowered (Figure F–5D).

It is natural to think that, with inhibitory firing rates lowered, excitatory firing rates should then rise back above their initial levels, but this does not occur. To understand this we must understand one more point about an unstable excitatory subnetwork. Recall that an increase in excitatory firing recruits so much extra recurrent excitation that, in the absence of changes in inhibitory firing, it would drive excitatory firing still higher. So too a decrease in excitatory firing withdraws so much recurrent excitation that excitatory firing rates would fall still lower in the absence of changes in inhibitory firing. The lowering of inhibitory firing rates decreases feedback inhibition, thus compensating for the deficiency of excitation and so stabilizing the lower firing rates of the excitatory population. The network thus arrives at a new stable fixed point in which both excitatory and inhibitory cells have lower firing rates than they did before additional external excitation was added to the inhibitory population.

This paradoxical result, in which adding excitatory input to the inhibitory cells results in a decrease in their steady state firing rate, is actually another instance of a hidden feed-forward connection by which a small differential or imbalance between excitation and inhibition can drive a large common response of both excitation and inhibition. In the present case the addition of excitatory input to the inhibitory cells drives a negative imbalance, which in turn drives a large negative common response, that is, a decrease in both excitatory and inhibitory rates. However, the increase in input also directly drives an increase in inhibitory firing rates. Thus there are two competing effects on inhibitory firing. The instability of the excitatory subnetwork turns out to be precisely equivalent to the condition in which the feed-forward effect is larger than the direct input effect, so that the net effect is a decrease in inhibitory firing rates.

The effect shown in Figure F–5 matches the observations we described earlier. When a stimulus in the receptive field surround results in additional excitatory input to inhibitory neurons, both the excitation and the inhibition in the recorded neurons is reduced. This finding and the theoretical analysis of it (only a portion of which is discussed here) provide strong evidence that the cortex operates in a regime in which recurrent excitation by itself is strong enough to be unstable but is stabilized by recurrent inhibition.

Recurrent Networks Can Model Decision-Making

As a final example of network modeling, we turn to circuits that select between different behaviors, that is, circuits that make decisions. Suppose the driver of an automobile needs to decide whether to turn right or left. Assume that there are two populations of excitatory neurons, one active when the decision is a right turn, the other when it is a left turn. Under some circumstances no decision needs to be made, so neither population should be highly active.

A decision can be biased or unbiased by sensory input. If, for example, the sensory stimulus is a road sign that says "turn left," the sensory input should bias the decision toward a left turn. If the sensory stimulus is an obstacle in the middle of a three-lane highway, there is a need to turn but the direction may be arbitrary. In this case the sensory input should evoke a decision without biasing it. Between these extremes, inputs may provide a range of biasing effects.

To model decision-making in this context we follow the work of X. J. Wang, who has modeled decision- making circuits extensively. A network model of the kind of decision-making described above must have the following properties. First, in the absence of relevant sensory stimuli there should be a stable pattern of spontaneous activity corresponding to no decision. Second, a sensory stimulus requiring a decision should eliminate or destabilize the no-decision state and introduce two new stable firing patterns corresponding to the two possible actions. Third, sensory stimuli should be capable of biasing the outcome so that one of these decision states is more likely to occur than the other.

In our model different decision states are represented by two recurrently connected networks of excitatory neurons, both of which excite a single population of inhibitory neurons that return feed-back inhibition to both of them (Figure F–6A). In the absence of sensory stimuli the network stays in a no-decision state in which the neurons have low activity. In this state we assume that the neurons have a low level of input, and their activity reflects an equilibrium involving this input, recurrent excitation, and feedback inhibition. A stimulus that induces a decision takes the form of excitatory drive to both excitatory populations of the network.

When there is no bias favoring one decision over the other, the inputs to the two excitatory populations are equal. Nevertheless, the model does make a decision—the firing rate of one population rises to and remains at a high level, while that of the other population falls back to a low firing rate after an initial rise (Figure F–6A). This occurs because the stimulus-generated inputs force both excitatory networks away from their no-decision firing rates. With the no-decision state eliminated, only two stable states remain available to the system, each corresponding to a different decision. One population ends up at a higher rate because small random fluctuations in the firing rates happen to favor that group (a small amount of noise was added to the model to generate these fluctuations). As a result of the fluctuations, each decision occurs 50% of the time.

When the stimulus is biased in favor of one population, the input to that population is higher than the input to the other. As a result, the firing rate of the favored population rises and remains high, while that of the other population falls after a brief and small rise (Figure F–6B). A strong-enough stimulus bias will produce the favored decision almost 100% of the time.

The latency from the time of the stimulus to the time a decision occurs can be determined by examining the divergence between the firing rates of the two neuronal populations. A decision is made more rapidly when the stimulus is biased than when it is unbiased (Figure F–6). This situation is similar to what is observed in the experiments on perceptual decision-making. In these experiments a monkey is trained to report the perceived direction of dots that move in many different directions on a screen. The monkey's performance depends on the coherence of motion of the dots. On most trials the dots have an overall tendency to move in one of two possible directions, introducing a type of stimulus bias. In some trials the coherence is zero, but the monkey still has to choose a direction of motion from these two possibilities. This situation is analogous to that shown in Figure F–6A.

There is experimental evidence that neural populations representing the two choice possibilities may be located in the posterior parietal cortex, specifically the lateral intraparietal area. Neurons recorded in this area by Michael Shadlen and collaborators during perceptual decision experiments involving moving dots behave like the model neurons in Figure F–6. Neural activity is initially driven by the sensory stimulus but then increases for one decision and decreases for the other. The final decision is preceded by a ramping activity that is relatively slow in the case of zero coherence of motion of the dots. This may be an indication of the existence of a period during which fluctuations are accumulating until a decision is made.

How does the model shown in Figure F–6 work and how was it constructed? The model relies on two key elements: bistability and inhibition-mediated competition between the two populations of excitatory neurons. Bistability is the ability of a network to sustain activity corresponding to either of two different states, typically one with a low rate of firing and one with a high rate. Persistent firing at a high rate is made possible by strong recurrent excitation. Bistability requires the level of activity of a neuron to depend on its recurrent input in a nonlinear way. In the example of Figure F–6 the nonlinearity is assured by a sigmoidal neuronal response curve that makes the firing rate relatively insensitive to changes in input both at low firing rates (when mean voltage is well below threshold) and at high firing rates (when rates saturate) but highly sensitive at intermediate firing rates. At an intermediate level of firing the high sensitivity to changes in input renders the excitatory feedback unstable, forcing the network to high or low firing rates. At high or low firing rates the excitatory feedback becomes stable because of the weakened neuronal sensitivity, allowing a stable firing rate.

In a bistable network low and high firing rates are stable in the presence of small transient input pulses. Even if these transient inputs modify the firing rate of the network, the firing rate returns to its initial state after the input pulse terminates. However, larger input pulses can induce transitions from one firing rate to the other (Figure F–7).

The other crucial component in the model is inhibition. The decision model in Figure F–6 consists of two populations of excitatory neurons, each of which can be bistable, and an inhibitory population that is driven by both excitatory populations and reciprocally inhibits both of them. The purpose of the inhibitory population is to avoid the state in which both excitatory populations fire at high rates, which would correspond to making both decisions at once. This state is avoided by introducing sufficiently strong inhibition of the two excitatory populations. The resulting model is described mathematically in Box F–3. It and models like it provide an elegant way of simulating decision making, including such features as the time required to make a decision and the frequency of errors.