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Gain Modulation
Gain Modulation 485
Gain Modulation E Salinas, Wake Forest University School of Medicine, Winston-Salem, NC, USA ã 2009 Elsevier Ltd. All rights reserved.
Origin and Key Issues The term ‘gain field’ was coined to describe an interaction between visually evoked responses and eye position first documented by Andersen and Mountcastle in 1983. In their experiment, as well as in those that followed, neurons from parietal cortex were recorded in awake monkeys and the responses were systematically mapped by varying both eye position and the location of a spot of light. Surprisingly, although the shape of the response curves as functions of stimulus location did not change with eye position, their amplitude, or gain, did. Figure 1(e) shows an idealized example. In analogy with the receptive field, the gain field corresponded to the full map showing how the response changed as a function of eye position. Initial work described gain fields that depended on eye position, but an additional dependence on head position was demonstrated later, such that the combination of head and eye angles – the gaze angle – seems to be the relevant quantity. These interactions are well approximated by a multiplication of two factors, such that the firing rate r of a visual parietal neuron can be expressed as r ¼ f(x)g(y), where x is the location of a visual stimulus and y the gaze angle. The functions f and g may have a variety of forms, but what is important computationally is that there are two separate terms that are combined nonlinearly (multiplicatively in this case). In the last 25 years or so, abundant experimental evidence has accumulated showing that, in the central nervous system, gain modulation is a widespread mechanism for combining information from multiple sources, not just vision and gaze angle. At the same time, complementary modeling studies have addressed two issues: (1) the kinds of computational problems that can be solved or substantially simplified by using gain modulation and (2), the biophysical mechanisms that may give rise to nonlinear interactions between two types of inputs converging on a neuron, as observed in experiments reporting changes in gain. Some of the highlights of this work are described here.
Gain Modulation and Coordinate Transformations Zipser and Andersen were the first to demonstrate the power of gain modulation in a specific biological
computation. They trained an artificial neural network to transform the location of a stimulus from an initial eye-centered representation to one in head-centered coordinates. Their network essentially learned to produce multiple neural responses like that in Figure 1(f) given an input that encoded gaze angle and a set of standard eye-centered (or retino-centered) responses like the one in Figure 1(d). What they found, after their network learned to perform the transformation, were intermediate model units that had developed gaze-dependent gain fields much like those in parietal cortex (see Figure 1(e)). This result suggested that gain fields were in some sense optimal for performing a coordinate transformation, which was an extraordinary breakthrough. However, it was not entirely clear why gain modulation was optimal. Subsequent theoretical studies showed that a network of neurons that respond to stumulus location x and are gain-modulated by gaze angle y is useful because it acts as a flexible set of building blocks from which any other, arbitrary functions of x and y can be constructed downstream. Mathematically, the gain-modulated neurons are said to serve as a basis set, meaning that by adding their responses in various proportions one can construct any desired function of x and y. Furthermore, with gain-modulated cells many important downstream functions can be generated using simple synaptic modification rules (Hebbian, or correlation-based, learning rules). The implication is that if area A has neurons that respond to stimulus x and are gain modulated by a quantity y, neurons in a downstream area B that are driven by A will probably respond in more complicated ways as functions of x and y; perhaps they will respond to xþy or xy, in which case their tuning curves as functions of x will shift with different values of y (as in Figure 1(f )), or perhaps the dependencies will be nonlinear. Tuning curve shifts associated with a variety of coordinate transformations have been documented in several cortical and subcortical areas, including the lateral parietal area (LIP), the ventral parietal area (VIP), premotor cortex, and the superior colliculus. Other Reference Frames
Although the variables x and y introduced here initially represented target location and eye position, the mechanism for generating transformed representations is extremely general, so the results are valid for any other encoded variables. Indeed, neurophysiological experiments have documented modulation by a variety of internal or proprioceptive signals. Examples include gaze direction, eye or head velocity, arm position, and attentional location. Typically, these signals seem to be directly combined with spatially tuned
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Figure 1 Transforming stimulus location from retino-centered to head-centered coordinates. The top panels schematize experiments in which visual receptive fields are mapped at three eye positions. The cross corresponds to the fixation point; small dots above it indicate locations tested with flashed stimuli. Blue and red circles indicate receptive fields of neurons in (d) and (f) respectively. Gaze angle is y. (a) Gaze is directed to the left. (b) Gaze is directed straight ahead. (c) Gaze is directed to the right. (d) Responses of a hypothetical early visual neuron in the three conditions schematized above. Firing rate is plotted as a function of stimulus location x, which is measured relative to the fixation point (i.e., in retinal coordinates). The tuning curve is independent of gaze angle. (e) Responses of a hypothetical parietal neuron that is gain modulated by gaze. Its receptive field location stays invariant relative to the fixation point, but its amplitude or gain depends on gaze direction. (f) Responses of a hypothetical neuron that encodes stimulus location in head-centered coordinates. Its receptive field stays in the same location relative to the head, so its tuning curve shifts as a function of x.
sensory responses to produce a change in reference frame, as in the case of eye position. The generality of this framework – the idea that arrays of gain-modulated neurons constitute basis sets for constructing more elaborate representations downstream – also extends the concept of coordinate transformation to perceptual processes, such as object recognition. For instance, directing attention to different locations in space modifies the gain of many visual neurons that react selectively to specific features of an image. According to work by Salinas and Abbott, such attentional modulation may give rise to visual responses that are, to a good degree, independent of where they appear in the visual field. Some neurons in inferior-temporal cortex (IT) seem to have this property, also known as translation invariance: they respond depending only on the type of image presented, regardless of its location. According to the theory, they encode visual information in attentioncentered coordinates. These cells are considered the neural correlate of our ability to recognize objects independently of where they appear in the visual field.
Partial Shifts in Recurrent Networks
Recurrent connections are ubiquitous in cortical networks, so it is important to understand how computations mediated by gain modulation change in networks with feedback interactions. Studies by Xing and Andersen and by Pouget and collaborators have investigated this in networks trained to perform coordinate transformations to and from diverse reference frames, but where feedback between neurons is allowed. With such feedback, the sensory responses that are optimal for implementing the transformations – the basis functions – still show nonlinear dependencies on the relevant proprioceptive parameters (eye position, head position, etc.). But in addition to changes in gain, their tuning curves show partial shifts as functions of stimulus location, particularly when more than two reference frames are involved. In other words, with feedback and multiple coordinate systems, the most efficient neuronal responses turn out to be intermediate between those shown in Figures 1(e) and 1(f), so individual neurons should display changes
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in gain together with tuning curve shifts that are smaller than those expected for a full transformation. Indeed, this mixture is typical of data from various experimental preparations, suggesting that models with feedback capture more of the complexity of real neuronal circuits.
Arbitrary Visuomotor Remapping Most theoretical studies on gain modulation have focused on the idea that proprioceptive signals combined with spatial sensory information lead to changes in reference frame. However, gain interactions may serve to integrate sensory information with other types of events, not necessarily proprioceptive, and in that case the mechanism works for establishing arbitrary associations between sensory stimuli and motor actions. This refers to tasks or behaviors in which a given stimulus is arbitrarily associated with two or more motor responses depending on separate cues, which provide a context. A simple example is illustrated in Figure 2, where the color of the fixation point indicates whether the correct movement should be a saccade (eye movement) toward a spot of light or an antisaccade away from it. Importantly, the stimulus may appear at many different locations. Gain modulation can be used to construct network models for tasks like this, in which multiple maps between sensory stimuli and motor actions are possible, but only one map, depending on the context, is implemented at any given time. This is illustrated in Figures 3 and 4.
unit has a preferred stimulus location and a preferred context, with red and green corresponding to saccade and antisaccade conditions, respectively. Modulation strength varies across cells and is strongest for the unit in Figure 3(a), which is fully suppressed in the nonpreferred context. Such a ‘switching’ neuron is entirely turned on and off by context. It is straightforward to see that an array of switching neurons can be easily configured to generate saccades in one context and antisaccades in the other: the neurons that are turned on in context 1 can establish their own set of synaptic connections regardless of the connections established by the other neurons, which are active only in context 2, and vice versa. However, to switch sensory–motor maps it is not necessary that individual neurons turn on and off with context. Figure 4 illustrates this point. The results in Figure 4 were obtained with a simple network without feedback in which 60 gainmodulated sensory neurons drive a set of 25 output or motor neurons through synaptic connections, and the output firing rates (black dots) are determined by weighted sums of the sensory firing rates (colored dots), such that X Ri ðx; yÞ ¼ wij rj ðx; yÞ ½1 j
40 Firing rate
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Figure 3 depicts four hypothetical tuning curves from neurons that respond in the task of Figure 2. Each
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Figure 2 An arbitrary sensory–motor mapping task. In each trial, a stimulus (black dot) is presented at a distance x from the fixation point (colored dot); the stimulus disappears; two targets appear (gray dots) and the subject responds by making an eye movement (arrow) to one of them. The color of the fixation spot acts as a contextual cue indicating whether the movement should be a saccade or an antisaccade. (a) In context 1 the fixation spot is red and the movement is to the target at x. (b) In context 2 the fixation spot is green and the movement is to the opposite target, at x. Adapted from Salinas E (2004) Context-dependent selection of visuomotor maps. BioMed Central Neuroscience 5: 47.
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Figure 3 Responses of four hypothetical gain-modulated neurons in the remapping task of Figure 2. Each graph plots the mean firing rate of a model neuron as a function of stimulus location. Red and green traces correspond to sensory responses evoked during contexts 1 and 2, respectively. (a) A unit that prefers context 1 and is 100% suppressed in the nonpreferred condition. (b) A unit that prefers context 2 and is 62% suppressed in the nonpreferred condition. (c) A unit that prefers context 1 and is 39% suppressed in context 2. (d) A unit that prefers context 2 and is 13% suppressed in context 1. Adapted from Salinas E (2004) Context-dependent selection of visuomotor maps. BioMed Central Neuroscience 5: 47.
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Figure 4 A model network that switches between saccades and antisaccades. The model consists of 60 sensory neurons driving 25 motor or output neurons through synaptic connections. The box marks a single trial. Colored traces are the firing rates of the 60 gain-modulated neurons in the network; 30 of them (red) prefer the direct saccade condition and 30 (green) prefer the antisaccade condition. Black dots are the 25 motor responses driven by the gain-modulated neurons. The x-axis is preferred stimulus location for the sensory cells and preferred movement location for the motor cells. In context 1 (direct saccades), the peak of motor activity should be aligned with the peak of sensory activity; in context 2 (antisaccades), the motor activity should peak at the mirror-symmetric point from the sensory activity. (a) Firing rates of all model cells when the sensory units are fully modulated by context (switching neurons). The context in each of four trials is indicated on the left. Trials with x ¼ 15 and x ¼ 10 alternate. The profile of output activity always peaks at the location of the correct saccade. (b) As in (a), but when all gain-modulated neurons are partially modulated by the same amount. (c) As in (a), but when all the gain factors are chosen randomly. Sets of neurons with diverse gain modulation factors can, with appropriate synaptic weights, perform the same input-output transformations as neurons that are fully modulated. Adapted from Salinas E (2004) Context-dependent selection of visuomotor maps. BioMed Central Neuroscience 5: 47.
Here, Ri(x, y) is the firing rate of output unit i, wij is the synaptic connection from sensory neuron j to output unit i, and rj(x, y) is the firing rate of sensory neuron j when the context is y (1 or 2) and the stimulus is located at x. In Figure 4(a) the model neurons are fully modulated by context, so only one subnetwork is active in context 1 (red dots) and only the other in context 2 (green dots). The two subnetworks are never active simultaneously, so they can establish independent connection matrices and generate separate sensory–motor maps for contexts 1 and 2. In Figure 4(b) the model works similarly, except that the sensory neurons are only partially suppressed in their nonpreferred context. Although the synaptic connections that allow the model to perform the two versions of the task (saccades and antisaccades) are different than in Figure 4(a), with the appropriate connections wij, performance is again accurate. This is true even when each sensory cell is modulated by a different, random amount, as in Figure 4(c). More generally, the key result is that, if certain mild conditions on the modulation strengths are satisfied,
then any downstream function Ri(x, y) that can be constructed with switching neurons can also be constructed with neurons that are only partially suppressed. In practice, this means that anything that can be done with subpopulations of switching neurons can also be done with neurons that modulate their responses only moderately. This result offers a useful intuition as to why gain modulation is so powerful: modulating the activity of a neuronal population is equivalent to flipping a switch that turns on or off various sets of neurons. Evidence of Context-Dependent Modulation
Although relatively little is known about how sensory-triggered activity is modified by arbitrary contextual cues, there are several experimental reports revealing (1) neuronal responses that are sensitive to more than one variable or stimulus feature and (2) that the interactions between those variables are generally nonlinear, and thus consistent with changes in gain. For example, the activity evoked in prefrontal neurons by moving dots varies substantially depending on
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whether their color or their direction of motion is relevant to the task at hand. Similarly, visual responses may change depending on the probability of obtaining a reward in the current trial, or on the rule that is used to link them to a motor behavior. Such effects could be used for generating abstract transformations in which the contextual or modulatory signal itself lacks any spatial information.
Mechanisms Underlying Changes in Gain What biophysical mechanisms might allow a neuron to combine two input signals in a multiplicative way? The problem is this. Suppose there are two nearby synapses, a and b, that generate depolarizations DVa and DVb, respectively, when activated individually. To a first approximation, the depolarization observed when activated together should be DVa þ DVb, and indeed this assumption is valid under various conditions. The question is how to make the combined response of synapses a and b nonlinear, and in particular, how to make it so that the neuron’s firing rate behaves approximately as a product of a- and b-dependent terms, as observed in extracellular recordings. Proposed Solutions
In recent years this issue has attracted considerable attention, so several candidate mechanisms with quite different properties have been proposed. One of the earliest suggestions was that of Mel, who showed using biophysically detailed computer simulations, that dendritic conductances with nonlinear voltage dependencies could boost the responses of nearby synapses that are activated simultaneously. That is, due to depolarizing currents in the dendrites, nearby synapses a and b that are active together can generate a depolarization that is much larger than DVa þ DVb. There is experimental support for this effect: on some pyramidal neurons, nearby synapses on the same dendritic branch appear to sum sigmoidally, whereas synapses that are either far apart or on different branches sum linearly. A second possibility is based on the observation that circuits with positive feedback tend to amplify their inputs. In this scheme, strong recurrent activity between neurons with similar gain fields is the basis for generating a multiplicative effect without the need for synapses to interact. Another mechanism that does not place any restrictions on the synapses themselves has been proposed by Chance and collaborators. They showed that the gain of a neuronal response to excitatory drive can be modulated by varying the overall level of synaptic input. Simultaneously increasing both excitatory and inhibitory background firing rates in a
balanced manner results in a divisive effect that modulates the responsiveness to excitatory drive. This, however, requires coordinated changes in the background activity; that is, although each input neuron needs to modify its firing rate by only a small amount, similar changes must occur across a large number of input neurons for the mechanism to work. In addition to these proposals, shunting inhibition, inhibitory synchrony, and input–output functions described by a power law have also been discussed in theoretical studies as being possibly implicated in regulating the gain of neurons. Two more mechanisms have been suggested recently on the basis of experimental observations. The first one is based on results by Larkum and colleagues showing that, in layer 5 pyramidal cells, there is a nonlinear interaction between inputs that arrive at the soma and at the apical dendrites. The key is that back-propagating action potentials help activate high-threshold dendritic calcium conductances, such that bursts of spikes are produced only when somatic and dendritic inputs coincide. Thus, a weak, asynchronous current at the distal dendrites can generate extra spikes, because of the evoked bursts, and this increases the gain of the response to somatic input. Finally, Mehaffey and colleagues suggested that a neuron’s gain may be controlled through inhibition that is applied at the dendrites, and specifically not at the soma. Due to the back-propagation of action potentials and to voltage-dependent currents at the dendrites, after a neuron fires a spike, a depolarizing afterpotential is typically generated. The crucial observation is that the size of this afterpotential regulates the gain of the cell, and that dendritic inhibition – but not somatic inhibition – reduces it. Therefore, dendritic inhibition decreases the gain. This last idea is interesting because the biophysical requirements for this form of divisive gain control are relatively simple. However, it is likely that neurons use a variety of mechanisms to regulate their gain over different timescales, so the possibilities mentioned earlier (and future ones) should not be considered mutually exclusive. See also: Motor Psychophysics; Neural Coding of Spatial Representations; Parietal Cortex and Spatial Attention; Saccadic Eye Movements; Spatial Transformations for Eye–Hand Coordination.
Further Reading Andersen RA, Essick GK, and Siegel RM (1985) Encoding of spatial location by posterior parietal neurons. Science 230: 450–458. Andersen RA and Mountcastle VB (1983) The influence of the angle of gaze upon the excitability of light-sensitive neurons
490 Gain Modulation of the posterior parietal cortex. Journal of Neuroscience 3: 532–548. Avillac M, Deneve S, Olivier E, Pouget A, and Duhamel JR (2005) Reference frames for representing visual and tactile locations in parietal cortex. Nature Neuroscience 8: 941–949. Brotchie PR, Andersen RA, Snyder LH, and Goodman SJ (1995) Head position signals used by parietal neurons to encode locations of visual stimuli. Nature 375: 232–235. Chance FS, Abbott LF, and Reyes AD (2002) Gain modulation from background synaptic input. Neuron 35: 773–782. Deneve S, Latham PE, and Pouget A (2001) Efficient computation and cue integration with noisy population codes. Nature Neuroscience 4: 826–831. Graziano MSA, Hu TX, and Gross CG (1997) Visuospatial properties of ventral premotor cortex. Journal of Neurophysiology 77: 2268–2292. Larkum ME, Senn W, and Lu¨scher H-R (2004) Top-down dendritic input increases the gain of layer 5 pyramidal neurons. Cerebral Cortex 14: 1059–1070. Lauwereyns J, Sakagami M, Tsutsui K, Kobayashi S, Koizumi M, and Hikosaka O (2001) Responses to task-irrelevant visual features by primate prefrontal neurons. Journal of Neurophysiology 86: 2001–2010. Mehaffey WH, Doiron B, Maler L, and Turner RW (2005) Deterministic multiplicative gain control with active dendrites. Journal of Neuroscience 25: 9968–9977. Poggio T (1990) A theory of how the brain might work. Cold Spring Harbor Symposium on Quantitative Biology 5: 899–910.
Polsky A, Mel BW, and Schiller J (2004) Computational subunits in thin dendrites of pyramidal cells. Nature Neuroscience 7: 621–627. Pouget A and Sejnowski TJ (1997) Spatial transformations in the parietal cortex using basis functions. Journal of Cognitive Neuroscience 9: 222–237. Salinas E (2004) Context-dependent selection of visuomotor maps. BioMed Central Neuroscience 5: 47. Salinas E and Abbott LF (1995) Transfer of coded information from sensory to motor networks. Journal of Neuroscience 15: 6461–6474. Salinas E and Abbott LF (1997) Invariant visual responses from attentional gain fields. Journal of Neurophysiology 77: 3267–3272. Salinas E and Their P (2000) Gain modulation: A major computational principle of the central nervous system. Neuron 27: 15–21. Wallis JD and Miller EK (2003) From rule to response: Neuronal processes in the premotor and prefrontal cortex. Journal of Neurophysiology 90: 1790–1806. Xing J and Andersen RA (2000) Models of the posterior parietal cortex which perform multimodal integration and represent space in several coordinate frames. Journal of Cognitive Neuroscience 12: 601–614. Zipser D and Andersen RA (1988) A back-propagation programmed network that simulates response properties of a subset of posterior parietal neurons. Nature 331: 679–684.
Gain Modulation E Salinas, Wake Forest University School of Medicine, Winston-Salem, NC, USA ã 2009 Elsevier Ltd. All rights reserved.
Origin and Key Issues The term ‘gain field’ was coined to describe an interaction between visually evoked responses and eye position first documented by Andersen and Mountcastle in 1983. In their experiment, as well as in those that followed, neurons from parietal cortex were recorded in awake monkeys and the responses were systematically mapped by varying both eye position and the location of a spot of light. Surprisingly, although the shape of the response curves as functions of stimulus location did not change with eye position, their amplitude, or gain, did. Figure 1(e) shows an idealized example. In analogy with the receptive field, the gain field corresponded to the full map showing how the response changed as a function of eye position. Initial work described gain fields that depended on eye position, but an additional dependence on head position was demonstrated later, such that the combination of head and eye angles – the gaze angle – seems to be the relevant quantity. These interactions are well approximated by a multiplication of two factors, such that the firing rate r of a visual parietal neuron can be expressed as r ¼ f(x)g(y), where x is the location of a visual stimulus and y the gaze angle. The functions f and g may have a variety of forms, but what is important computationally is that there are two separate terms that are combined nonlinearly (multiplicatively in this case). In the last 25 years or so, abundant experimental evidence has accumulated showing that, in the central nervous system, gain modulation is a widespread mechanism for combining information from multiple sources, not just vision and gaze angle. At the same time, complementary modeling studies have addressed two issues: (1) the kinds of computational problems that can be solved or substantially simplified by using gain modulation and (2), the biophysical mechanisms that may give rise to nonlinear interactions between two types of inputs converging on a neuron, as observed in experiments reporting changes in gain. Some of the highlights of this work are described here.
Gain Modulation and Coordinate Transformations Zipser and Andersen were the first to demonstrate the power of gain modulation in a specific biological
computation. They trained an artificial neural network to transform the location of a stimulus from an initial eye-centered representation to one in head-centered coordinates. Their network essentially learned to produce multiple neural responses like that in Figure 1(f) given an input that encoded gaze angle and a set of standard eye-centered (or retino-centered) responses like the one in Figure 1(d). What they found, after their network learned to perform the transformation, were intermediate model units that had developed gaze-dependent gain fields much like those in parietal cortex (see Figure 1(e)). This result suggested that gain fields were in some sense optimal for performing a coordinate transformation, which was an extraordinary breakthrough. However, it was not entirely clear why gain modulation was optimal. Subsequent theoretical studies showed that a network of neurons that respond to stumulus location x and are gain-modulated by gaze angle y is useful because it acts as a flexible set of building blocks from which any other, arbitrary functions of x and y can be constructed downstream. Mathematically, the gain-modulated neurons are said to serve as a basis set, meaning that by adding their responses in various proportions one can construct any desired function of x and y. Furthermore, with gain-modulated cells many important downstream functions can be generated using simple synaptic modification rules (Hebbian, or correlation-based, learning rules). The implication is that if area A has neurons that respond to stimulus x and are gain modulated by a quantity y, neurons in a downstream area B that are driven by A will probably respond in more complicated ways as functions of x and y; perhaps they will respond to xþy or xy, in which case their tuning curves as functions of x will shift with different values of y (as in Figure 1(f )), or perhaps the dependencies will be nonlinear. Tuning curve shifts associated with a variety of coordinate transformations have been documented in several cortical and subcortical areas, including the lateral parietal area (LIP), the ventral parietal area (VIP), premotor cortex, and the superior colliculus. Other Reference Frames
Although the variables x and y introduced here initially represented target location and eye position, the mechanism for generating transformed representations is extremely general, so the results are valid for any other encoded variables. Indeed, neurophysiological experiments have documented modulation by a variety of internal or proprioceptive signals. Examples include gaze direction, eye or head velocity, arm position, and attentional location. Typically, these signals seem to be directly combined with spatially tuned
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Figure 1 Transforming stimulus location from retino-centered to head-centered coordinates. The top panels schematize experiments in which visual receptive fields are mapped at three eye positions. The cross corresponds to the fixation point; small dots above it indicate locations tested with flashed stimuli. Blue and red circles indicate receptive fields of neurons in (d) and (f) respectively. Gaze angle is y. (a) Gaze is directed to the left. (b) Gaze is directed straight ahead. (c) Gaze is directed to the right. (d) Responses of a hypothetical early visual neuron in the three conditions schematized above. Firing rate is plotted as a function of stimulus location x, which is measured relative to the fixation point (i.e., in retinal coordinates). The tuning curve is independent of gaze angle. (e) Responses of a hypothetical parietal neuron that is gain modulated by gaze. Its receptive field location stays invariant relative to the fixation point, but its amplitude or gain depends on gaze direction. (f) Responses of a hypothetical neuron that encodes stimulus location in head-centered coordinates. Its receptive field stays in the same location relative to the head, so its tuning curve shifts as a function of x.
sensory responses to produce a change in reference frame, as in the case of eye position. The generality of this framework – the idea that arrays of gain-modulated neurons constitute basis sets for constructing more elaborate representations downstream – also extends the concept of coordinate transformation to perceptual processes, such as object recognition. For instance, directing attention to different locations in space modifies the gain of many visual neurons that react selectively to specific features of an image. According to work by Salinas and Abbott, such attentional modulation may give rise to visual responses that are, to a good degree, independent of where they appear in the visual field. Some neurons in inferior-temporal cortex (IT) seem to have this property, also known as translation invariance: they respond depending only on the type of image presented, regardless of its location. According to the theory, they encode visual information in attentioncentered coordinates. These cells are considered the neural correlate of our ability to recognize objects independently of where they appear in the visual field.
Partial Shifts in Recurrent Networks
Recurrent connections are ubiquitous in cortical networks, so it is important to understand how computations mediated by gain modulation change in networks with feedback interactions. Studies by Xing and Andersen and by Pouget and collaborators have investigated this in networks trained to perform coordinate transformations to and from diverse reference frames, but where feedback between neurons is allowed. With such feedback, the sensory responses that are optimal for implementing the transformations – the basis functions – still show nonlinear dependencies on the relevant proprioceptive parameters (eye position, head position, etc.). But in addition to changes in gain, their tuning curves show partial shifts as functions of stimulus location, particularly when more than two reference frames are involved. In other words, with feedback and multiple coordinate systems, the most efficient neuronal responses turn out to be intermediate between those shown in Figures 1(e) and 1(f), so individual neurons should display changes
Gain Modulation 487
in gain together with tuning curve shifts that are smaller than those expected for a full transformation. Indeed, this mixture is typical of data from various experimental preparations, suggesting that models with feedback capture more of the complexity of real neuronal circuits.
Arbitrary Visuomotor Remapping Most theoretical studies on gain modulation have focused on the idea that proprioceptive signals combined with spatial sensory information lead to changes in reference frame. However, gain interactions may serve to integrate sensory information with other types of events, not necessarily proprioceptive, and in that case the mechanism works for establishing arbitrary associations between sensory stimuli and motor actions. This refers to tasks or behaviors in which a given stimulus is arbitrarily associated with two or more motor responses depending on separate cues, which provide a context. A simple example is illustrated in Figure 2, where the color of the fixation point indicates whether the correct movement should be a saccade (eye movement) toward a spot of light or an antisaccade away from it. Importantly, the stimulus may appear at many different locations. Gain modulation can be used to construct network models for tasks like this, in which multiple maps between sensory stimuli and motor actions are possible, but only one map, depending on the context, is implemented at any given time. This is illustrated in Figures 3 and 4.
unit has a preferred stimulus location and a preferred context, with red and green corresponding to saccade and antisaccade conditions, respectively. Modulation strength varies across cells and is strongest for the unit in Figure 3(a), which is fully suppressed in the nonpreferred context. Such a ‘switching’ neuron is entirely turned on and off by context. It is straightforward to see that an array of switching neurons can be easily configured to generate saccades in one context and antisaccades in the other: the neurons that are turned on in context 1 can establish their own set of synaptic connections regardless of the connections established by the other neurons, which are active only in context 2, and vice versa. However, to switch sensory–motor maps it is not necessary that individual neurons turn on and off with context. Figure 4 illustrates this point. The results in Figure 4 were obtained with a simple network without feedback in which 60 gainmodulated sensory neurons drive a set of 25 output or motor neurons through synaptic connections, and the output firing rates (black dots) are determined by weighted sums of the sensory firing rates (colored dots), such that X Ri ðx; yÞ ¼ wij rj ðx; yÞ ½1 j
40 Firing rate
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Figure 3 depicts four hypothetical tuning curves from neurons that respond in the task of Figure 2. Each
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Figure 2 An arbitrary sensory–motor mapping task. In each trial, a stimulus (black dot) is presented at a distance x from the fixation point (colored dot); the stimulus disappears; two targets appear (gray dots) and the subject responds by making an eye movement (arrow) to one of them. The color of the fixation spot acts as a contextual cue indicating whether the movement should be a saccade or an antisaccade. (a) In context 1 the fixation spot is red and the movement is to the target at x. (b) In context 2 the fixation spot is green and the movement is to the opposite target, at x. Adapted from Salinas E (2004) Context-dependent selection of visuomotor maps. BioMed Central Neuroscience 5: 47.
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Figure 3 Responses of four hypothetical gain-modulated neurons in the remapping task of Figure 2. Each graph plots the mean firing rate of a model neuron as a function of stimulus location. Red and green traces correspond to sensory responses evoked during contexts 1 and 2, respectively. (a) A unit that prefers context 1 and is 100% suppressed in the nonpreferred condition. (b) A unit that prefers context 2 and is 62% suppressed in the nonpreferred condition. (c) A unit that prefers context 1 and is 39% suppressed in context 2. (d) A unit that prefers context 2 and is 13% suppressed in context 1. Adapted from Salinas E (2004) Context-dependent selection of visuomotor maps. BioMed Central Neuroscience 5: 47.
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Figure 4 A model network that switches between saccades and antisaccades. The model consists of 60 sensory neurons driving 25 motor or output neurons through synaptic connections. The box marks a single trial. Colored traces are the firing rates of the 60 gain-modulated neurons in the network; 30 of them (red) prefer the direct saccade condition and 30 (green) prefer the antisaccade condition. Black dots are the 25 motor responses driven by the gain-modulated neurons. The x-axis is preferred stimulus location for the sensory cells and preferred movement location for the motor cells. In context 1 (direct saccades), the peak of motor activity should be aligned with the peak of sensory activity; in context 2 (antisaccades), the motor activity should peak at the mirror-symmetric point from the sensory activity. (a) Firing rates of all model cells when the sensory units are fully modulated by context (switching neurons). The context in each of four trials is indicated on the left. Trials with x ¼ 15 and x ¼ 10 alternate. The profile of output activity always peaks at the location of the correct saccade. (b) As in (a), but when all gain-modulated neurons are partially modulated by the same amount. (c) As in (a), but when all the gain factors are chosen randomly. Sets of neurons with diverse gain modulation factors can, with appropriate synaptic weights, perform the same input-output transformations as neurons that are fully modulated. Adapted from Salinas E (2004) Context-dependent selection of visuomotor maps. BioMed Central Neuroscience 5: 47.
Here, Ri(x, y) is the firing rate of output unit i, wij is the synaptic connection from sensory neuron j to output unit i, and rj(x, y) is the firing rate of sensory neuron j when the context is y (1 or 2) and the stimulus is located at x. In Figure 4(a) the model neurons are fully modulated by context, so only one subnetwork is active in context 1 (red dots) and only the other in context 2 (green dots). The two subnetworks are never active simultaneously, so they can establish independent connection matrices and generate separate sensory–motor maps for contexts 1 and 2. In Figure 4(b) the model works similarly, except that the sensory neurons are only partially suppressed in their nonpreferred context. Although the synaptic connections that allow the model to perform the two versions of the task (saccades and antisaccades) are different than in Figure 4(a), with the appropriate connections wij, performance is again accurate. This is true even when each sensory cell is modulated by a different, random amount, as in Figure 4(c). More generally, the key result is that, if certain mild conditions on the modulation strengths are satisfied,
then any downstream function Ri(x, y) that can be constructed with switching neurons can also be constructed with neurons that are only partially suppressed. In practice, this means that anything that can be done with subpopulations of switching neurons can also be done with neurons that modulate their responses only moderately. This result offers a useful intuition as to why gain modulation is so powerful: modulating the activity of a neuronal population is equivalent to flipping a switch that turns on or off various sets of neurons. Evidence of Context-Dependent Modulation
Although relatively little is known about how sensory-triggered activity is modified by arbitrary contextual cues, there are several experimental reports revealing (1) neuronal responses that are sensitive to more than one variable or stimulus feature and (2) that the interactions between those variables are generally nonlinear, and thus consistent with changes in gain. For example, the activity evoked in prefrontal neurons by moving dots varies substantially depending on
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whether their color or their direction of motion is relevant to the task at hand. Similarly, visual responses may change depending on the probability of obtaining a reward in the current trial, or on the rule that is used to link them to a motor behavior. Such effects could be used for generating abstract transformations in which the contextual or modulatory signal itself lacks any spatial information.
Mechanisms Underlying Changes in Gain What biophysical mechanisms might allow a neuron to combine two input signals in a multiplicative way? The problem is this. Suppose there are two nearby synapses, a and b, that generate depolarizations DVa and DVb, respectively, when activated individually. To a first approximation, the depolarization observed when activated together should be DVa þ DVb, and indeed this assumption is valid under various conditions. The question is how to make the combined response of synapses a and b nonlinear, and in particular, how to make it so that the neuron’s firing rate behaves approximately as a product of a- and b-dependent terms, as observed in extracellular recordings. Proposed Solutions
In recent years this issue has attracted considerable attention, so several candidate mechanisms with quite different properties have been proposed. One of the earliest suggestions was that of Mel, who showed using biophysically detailed computer simulations, that dendritic conductances with nonlinear voltage dependencies could boost the responses of nearby synapses that are activated simultaneously. That is, due to depolarizing currents in the dendrites, nearby synapses a and b that are active together can generate a depolarization that is much larger than DVa þ DVb. There is experimental support for this effect: on some pyramidal neurons, nearby synapses on the same dendritic branch appear to sum sigmoidally, whereas synapses that are either far apart or on different branches sum linearly. A second possibility is based on the observation that circuits with positive feedback tend to amplify their inputs. In this scheme, strong recurrent activity between neurons with similar gain fields is the basis for generating a multiplicative effect without the need for synapses to interact. Another mechanism that does not place any restrictions on the synapses themselves has been proposed by Chance and collaborators. They showed that the gain of a neuronal response to excitatory drive can be modulated by varying the overall level of synaptic input. Simultaneously increasing both excitatory and inhibitory background firing rates in a
balanced manner results in a divisive effect that modulates the responsiveness to excitatory drive. This, however, requires coordinated changes in the background activity; that is, although each input neuron needs to modify its firing rate by only a small amount, similar changes must occur across a large number of input neurons for the mechanism to work. In addition to these proposals, shunting inhibition, inhibitory synchrony, and input–output functions described by a power law have also been discussed in theoretical studies as being possibly implicated in regulating the gain of neurons. Two more mechanisms have been suggested recently on the basis of experimental observations. The first one is based on results by Larkum and colleagues showing that, in layer 5 pyramidal cells, there is a nonlinear interaction between inputs that arrive at the soma and at the apical dendrites. The key is that back-propagating action potentials help activate high-threshold dendritic calcium conductances, such that bursts of spikes are produced only when somatic and dendritic inputs coincide. Thus, a weak, asynchronous current at the distal dendrites can generate extra spikes, because of the evoked bursts, and this increases the gain of the response to somatic input. Finally, Mehaffey and colleagues suggested that a neuron’s gain may be controlled through inhibition that is applied at the dendrites, and specifically not at the soma. Due to the back-propagation of action potentials and to voltage-dependent currents at the dendrites, after a neuron fires a spike, a depolarizing afterpotential is typically generated. The crucial observation is that the size of this afterpotential regulates the gain of the cell, and that dendritic inhibition – but not somatic inhibition – reduces it. Therefore, dendritic inhibition decreases the gain. This last idea is interesting because the biophysical requirements for this form of divisive gain control are relatively simple. However, it is likely that neurons use a variety of mechanisms to regulate their gain over different timescales, so the possibilities mentioned earlier (and future ones) should not be considered mutually exclusive. See also: Motor Psychophysics; Neural Coding of Spatial Representations; Parietal Cortex and Spatial Attention; Saccadic Eye Movements; Spatial Transformations for Eye–Hand Coordination.
Further Reading Andersen RA, Essick GK, and Siegel RM (1985) Encoding of spatial location by posterior parietal neurons. Science 230: 450–458. Andersen RA and Mountcastle VB (1983) The influence of the angle of gaze upon the excitability of light-sensitive neurons
490 Gain Modulation of the posterior parietal cortex. Journal of Neuroscience 3: 532–548. Avillac M, Deneve S, Olivier E, Pouget A, and Duhamel JR (2005) Reference frames for representing visual and tactile locations in parietal cortex. Nature Neuroscience 8: 941–949. Brotchie PR, Andersen RA, Snyder LH, and Goodman SJ (1995) Head position signals used by parietal neurons to encode locations of visual stimuli. Nature 375: 232–235. Chance FS, Abbott LF, and Reyes AD (2002) Gain modulation from background synaptic input. Neuron 35: 773–782. Deneve S, Latham PE, and Pouget A (2001) Efficient computation and cue integration with noisy population codes. Nature Neuroscience 4: 826–831. Graziano MSA, Hu TX, and Gross CG (1997) Visuospatial properties of ventral premotor cortex. Journal of Neurophysiology 77: 2268–2292. Larkum ME, Senn W, and Lu¨scher H-R (2004) Top-down dendritic input increases the gain of layer 5 pyramidal neurons. Cerebral Cortex 14: 1059–1070. Lauwereyns J, Sakagami M, Tsutsui K, Kobayashi S, Koizumi M, and Hikosaka O (2001) Responses to task-irrelevant visual features by primate prefrontal neurons. Journal of Neurophysiology 86: 2001–2010. Mehaffey WH, Doiron B, Maler L, and Turner RW (2005) Deterministic multiplicative gain control with active dendrites. Journal of Neuroscience 25: 9968–9977. Poggio T (1990) A theory of how the brain might work. Cold Spring Harbor Symposium on Quantitative Biology 5: 899–910.
Polsky A, Mel BW, and Schiller J (2004) Computational subunits in thin dendrites of pyramidal cells. Nature Neuroscience 7: 621–627. Pouget A and Sejnowski TJ (1997) Spatial transformations in the parietal cortex using basis functions. Journal of Cognitive Neuroscience 9: 222–237. Salinas E (2004) Context-dependent selection of visuomotor maps. BioMed Central Neuroscience 5: 47. Salinas E and Abbott LF (1995) Transfer of coded information from sensory to motor networks. Journal of Neuroscience 15: 6461–6474. Salinas E and Abbott LF (1997) Invariant visual responses from attentional gain fields. Journal of Neurophysiology 77: 3267–3272. Salinas E and Their P (2000) Gain modulation: A major computational principle of the central nervous system. Neuron 27: 15–21. Wallis JD and Miller EK (2003) From rule to response: Neuronal processes in the premotor and prefrontal cortex. Journal of Neurophysiology 90: 1790–1806. Xing J and Andersen RA (2000) Models of the posterior parietal cortex which perform multimodal integration and represent space in several coordinate frames. Journal of Cognitive Neuroscience 12: 601–614. Zipser D and Andersen RA (1988) A back-propagation programmed network that simulates response properties of a subset of posterior parietal neurons. Nature 331: 679–684.