Synaptic Depression and Facilitation Can Induce Motion Aftereffects in an Excitable Membrane Model of Visual Motion Processing Davis Barch and Donald A. Glaser Graduate Program in Vision Science, and Department of Physics and of Neurobiology in the Graduate School 221 Donner Hall University of California at Berkeley Berkeley, CA 94720 [email protected] [email protected]

Abstract A two-dimensional sheet of locally connected neural elements may be used to model motion detection in visual cortex. Activity waves resulting from inputs representing objects in motion have distinctive amplitudes and shapes, which characterize the object's motion. Repeated application of moving stimuli to such a system may lead to depression and facilitation of connections in the membrane. As a consequence, nonmoving stimuli may then give rise to activity waves like those characteristic of motion, but in the direction opposite to that of the conditioning stimulus. This effect may be used to model translational and rotational motion aftereffects in visual cortex. Keywords: travelling activity waves; excitable membranes; motion detection; motion afterefects; neural models

Introduction Motion aftereffects are well known and well characterized phenomena associated with the perception of visual motion (see, for example, [6], and the references therein). The existence and nature of these aftereffects is of considerable interest as a means of studying the mechanisms underlying visual motion processing as well as for the role they may play in maintaining the effectiveness of human vision. We have demonstrated previously [3] that the response properties of an excitable membrane, modeled as a two-dimensional array of linear threshold units, can be used to extract and characterize motion signals presented as input to the membrane. The nature and interactions of traveling waves of activity in the membrane, in particular, can be used to model the psychophysics of motion sensing by biological visual systems. We demonstrate here that this model of motion processing exhibits motion aftereffects, if the efficacy of the connections between units is allowed to vary with the history of the activity passing through the connection. The identical architecture can give rise to translational, rotational, and other motion aftereffects. This mechanism may be used as a model for such aftereffects in the primate visual system.

The Model The model presented here is an excitable membrane, implemented as a two-dimensional sheet of connected neural elements, laid out in a hexagonal array. Units are linear threshold units (whose activity is bound by an upper limit and a lower limit (typically 0)); and unit activity levels are continuous. Input

to a unit may be externally applied, or may arise from connections with neighboring units. Unit activity at each time step is the activity at the previous time step, exponentially decayed, plus the sum of the inputs to the unit (multiplied by a synaptic efficiency factor). The architecture and governing equations of this model have been described previously [2,3]. We describe here an extension of this model, which includes depression and facilitation of each connection between two units, based on the history of the activity that has passed through the connection. As described above, each excitable membrane unit has a unit-level efficiency or gain. In this extension of the model, each connection has associated with it a connection efficacy. This efficacy is initially 1.00 for all connections, and is updated at each timestep, for each connection. Input to a given unit, at each timestep, is the sum of the activities (at the previous timestep) of the units from which the unit in question receives activity, times the strength of the connections from the neighboring units to the unit in question, times the efficacy of each connection. This is described by equation 1. I(x,y,t) = Σ(u,v) [ A(u,v, t) * W(u,v,x,y) * SE(u,v,x,y,t) ] + U(x,y,t)

(1),

where I (x,y,t) is the total input at time t to the unit at location (x,y); A(u,v, t) is the activity at time t of unit at location (u,v); W is the strength (weight) of the connection from the unit at (u,v) to the unit at (x,y); U(x,y,t) is input at time applied externally to the unit at location (x,y) at time t; and SE is the efficacy of the connection from the unit at location (u,v) to the unit at location (x,y) at time t. At each timestep the connection efficacy is acted on by three different factors: First, the efficacy of each connection tends to decay (upward or downward) towards 1.00. This tendency is given by the Efficacy Decay factor, derived in equation 2. ED(x,y,u,v,t+1) = 1 - [(1- SE(x,y,u,v,t)) * exp(sed*∆t ]

(2),

where ED(x,y,u,v,t) is the efficacy decay factor at time t+1 for the connection from the unit located at position (x,y) to the unit located at position (u,v); SE (x,y,u,v,t) is the synaptic efficacy of this connection at time t; sed is the decay (or recovery) rate for synaptic efficacy (this is constant for all units and all timesteps); and ∆t is the length of each time step (defined to be 1). Second, the efficacy of the connection from a source unit to a target unit is decreased (depressed) if the source unit is active at the current timestep, if the activity of the source unit increased in the previous timestep, and if the activity of the target unit increased in the current timestep. This is implemented as a tendency to decrease the difference between the current synaptic efficacy and the minimum possible efficacy. The amount by which this difference is decreased is given by the synaptic depression factor for this connection, which is governed by equation 3. SD(x,y,u,v,t) = exp[ Anorm(x,y,t) * ∆Anorm(x,y,t-1) * ∆Αnorm(u,v,t) * D)]

(3),

where SD(x,y,u,v,t) is the synaptic depression factor for the connection from the unit at location (x,y) to the unit at location (u,v) at time t; A(x,y,t) is the activity of the unit at position (x,y) at time t; Anorm (x,y,t) is A(x,y,t)/ Amax, where Amax is the maximum unit activity (typically 20 arbitrary activity units); and D is the synaptic efficacy depression parameter. In the absence of any of any other factors, the new synaptic efficacy would therefore be given by equation 4. SEdepress(x,y,u,v,t+1) = [ (SE(x,y,u,v,t) - SEmin ) * SD(x,y,u,v,t)] + SEmin.

(4),

where SE(x,y,u,v,t) is the current efficacy of the connection from the unit at location (x,y) to the unit at location (u,v) at time t; and SEmin is the minimum possible efficacy (this is constant over all connections and timesteps). The tendency of activity passing through a connection to depress the efficacy of the connection is implemented by deriving the ratio of the SEdepress factor for this connection to the current efficacy of the connection. This Synaptic Depression Ratio is given by equation 5. SDR(x,y,u,v,t+1) = SE(x,y,u,v,t) / SEdepress(x,y,u,v,t+1)

(5),

where SDR(x,y,u,v,t+1) is the synaptic depression ratio for the connection from the unit at location (x,y) to the unit at location (u,v) at time t+1. The third factor acting upon the efficacy of a connection from a source unit to a target unit is the tendency to facilitate the connection (increase the connection’s efficacy) if the source unit is active in the current timestep, if the activity of the source unit increased in the previous timestep, and if the activity of the target unit increased in the timestep prior to that (t-2). This facilitation is implemented as a tendency to decrease the difference between the current synaptic efficacy and the maximum possible efficacy. The amount by which this difference is decreased is given by the synaptic facilitation factor for this connection, which is governed by equation 6. SF(x,y,u,v,t) = exp [Anorm (x,y,t) * ∆Anorm (x,y,t-1) * ∆Αnorm(u,v,t-2) * F)]

(6),

where SF(x,y,u,v,t+1) is the synaptic facilitation factor for the connection from the unit at location (x,y) to the unit at location (u,v) at time t; A(x,y,t) is the activity of the unit at position (x,y) at time t; ANORM(x,y,t) is A(x,y,t)/ Amax, where Amax is the maximum unit activity (typically 20 arbitrary activity units); and F is the synaptic efficacy facilitation parameter. In the absence of any of any other factors, the new synaptic efficacy would therefore be given by equation 7. SEfacilitate(x,y,u,v,t+1) = SEmax – [ (SEmax - SE(x,y,u,v,t) ) * SF(x,y,u,v,t)]

(7),

where SE(x,y,u,v,t) is the current efficacy of the connection from the unit at location (x,y) to the unit at location (u,v) at time t; and SEmax is the maximum possible efficacy (this is constant over all connections and timesteps). The tendency of activity passing through a connection to facilitate the efficacy of the connection is implemented by deriving the ratio of the SEfacilitate factor for this connection to the current efficacy of the connection. This Synaptic Facilitation Ratio is given by equation 8. SFR(x,y,u,v,t+1) = SE(x,y,u,v,t) / SEfacilitate(x,y,u,v,t+1)

(8),

where SFR(x,y,u,v,t+1) is the synaptic facilitation ratio for the connection from the unit at location (x,y) to the unit at location (u,v) at time t+1. The efficacy of a connection is the product of the decay, depression, and facilitation factors derived above, and is given by equation 9. SE(x,y,u,v,t+1) = ED(x,y,u,v,t+1) * SDR(x,y,u,v,t+1) * SFR(x,y,u,v,t+1)

(9)

Results In a membrane such as the one described here, activity applied externally to a single location leads to an expanding circular activity wave, which may die away quickly, decay or grow slowly, or grow quickly, depending on the parameters of the membrane. If an external activity is applied to a series of units lying on the same straight line over a series of time steps (e.g., activity which is undergoing coherent linear motion), the resulting activity wave enjoys both a heightened amplitude and a characteristic "bow wave" shape if the speed of motion exceeds a characteristic propagation speed of the membrane. The existence of such an activity wave is therefore diagnostic of the presence of a motion signal in the input to the membrane, and may be used to characterize the speed and direction of the motion. Such a response is elicited from individual points or extended objects with similar motions, as can be seen in figure 1 for moving isolated points. Insert Figure 1 Here

An object undergoing a rotation gives rise to a characteristic curved activity wave, diagnostic of rotational motion. This effect can be seen in figure 2. Insert Figure 2 Here

As a stimulus moves across the membrane, the synaptic efficacy update rules described in equation 1 cause the efficacies of the connections in the direction of motion to be depressed, while connections in the opposite direction are facilitated. If the moving stimulus is applied repeatedly, and the rate with which connection efficacies recover is sufficiently slow, this effect may be cumulative and lasting. As a result, static stimuli presented to regions which have been conditioned by repeated application of moving stimuli may give rise to activity waves which quickly die away in the direction of the moving stimuli, but which persist and propagate in the opposite direction. In an unconditioned membrane, the existence of such activity waves indicates the presence of a moving stimulus. This effect is therefore called a motion aftereffect, as can be seen in figure 3.

Insert Figure 3 Here

If the conditioning stimulus is undergoing rotational motion, each region of the excitable membrane will become conditioned to motion in a slightly different direction. A static stimulus presented after such conditioning therefore gives rise to activity waves which continually change direction as they propagate across the membrane. This effect represents a rotational aftereffect, and can be seen in figure 4.

Insert Figure 4 Here

The existence of synaptic depression and facilitation in this excitable membrane model of motion detection can therefore give rise to both translational and rotational motion aftereffects, in the direction opposite to that of a conditioning stimulus.

Discussion The model described here is based on the premise that short-term synaptic depression is, roughly, the mirror image of long-term (hebbian) synaptic potentiation. It is similar in behavior, but opposite in sign (and of much shorter duration). Abbott et. al. ([5]) have demonstrated that measured synaptic depression in layer 3 of rat area V1 is best matched by models that allow for 2 or 3 different depression terms, with different sensitivities and recovery periods, as well as at least one facilitation term. The recovery time constants found in this study are as long as 8.5 seconds, comparable to the duration of visual aftereffects. We therefore allowed for both synaptic depression and facilitation in our model, but limited ourselves to a single timecourse for each. Recently, it has been demonstrated that the strength, and sign, of synapse-level long term potentiation and depression (LTP and LTD) depends on the relative timing of the spiking behavior of the pre- and postsynaptic neurons (see references [1] and [4], and the references therein). If a presynaptic neuron shares a synapse with a postsynaptic neuron, and the presynaptic neuron generates an action potential before the postsynaptic neuron, then the synapse is strengthened; if the presynaptic neuron spikes after the postsynaptic neuron, the synaptic weight decreases. The magnitude of this long-term synaptic potentiation or depression increases as the timing difference between the pre- and postsynaptic spikes decreases (this necessitates a discontinuity when the pre- and postsynaptic elements spike simultaneously, and the resultant synaptic effect crosses over from maximal potentiation to maximal depression.) The model we use here does not use spiking elements, but rather uses elements connected in continuous, analog fashion. At each timestep, each unit sends some nonnegative, continuously valued activity to all of the units to which it is connected (18 neighbors, for the results shown here). This may be taken to represent neurons that share analog (nonspiking) connections, or as modeling clusters of spiking neurons as a single analog element. In this latter interpretation of the model, the analog activity levels transmitted through a connection between two units represents the total number of spikes generated from a cluster of spiking elements in a given timestep. Based on this excitable membrane model, the learning rules described above, and our initial premise concerning short-term depression and long-term potentiation, we defined the following rule for shortterm depression and facilitation: if the source unit of a connection increases its activity level, and the target unit of that connection increases its activity at a later timestep (possibly due to activity from the source unit), the connection efficacy is depressed; if the source unit of a connection increases its activity level, and the target unit of that connection increased its activity at an earlier timestep the connection efficacy is facilitated. For simplicity (and computational tractability), we have limited this effect to a single timestep before and after the activity change of a given source unit. We have previously demonstrated ([3]) that an excitable membrane model of the visual cortex can respond to moving stimuli by generating characteristic activity waves, and that such activity waves can be used to detect and characterize translational and rotational motion. Extension of this model to include connection-level efficacies, and short-term changes in these efficacies based on the history of the activity which has passed through each connection, leads to extended (but temporary) changes in the behavior of the membrane in response to a conditioning stimulus. After extended exposure to such a conditioning stimulus, activity waves normally associated with moving stimuli are generated in response

to static stimuli, and are characteristic of motion in the direction opposite to that of the conditioning stimulus. We show here that a single mechanism can give rise to both translational and rotational motion aftereffects. It is expected that this mechanism can also give rise to any arbitrary motion aftereffect, depending only on the nature of the conditioning stimulus. There is an extensive literature regarding motion aftereffects (see, for example, [6]). We believe that comparison of the results described in that literature with the properties of the model we describe here will show that many of these results may be modeled in a simple and direct fashion by such a mechanism. This work remains to be done.

References [1] L.F. Abbott, S.B. Nelson., Synaptic plasticity: taming the beast, Nature Neuroscience 3 Suppl (2000) 1178-83 [2] D. Barch, Characterization of Activity Oscillations in an Excitable Membrane Model and Their Potential Functionality for Neuronal Computations, Neurocomputing 32-33 (2000) 25-35 [3] D.A. Glaser, D. Barch, Motion detection and characterization by an excitable membrane: the "Bow Wave" model, Neurocomputing 26-27 (1999) 137-46 [4] S. Song , K.D. Miller, L.F. Abbott,Competitive Hebbian learning through spiketiming-dependent synaptic plasticity, Nature Neuroscience 3 (9) (2000) 919-26 [5] J.A. Varela, K. Sen, J. Gibson, J. Fost, L.F. Abbott, S.B. Nelson, A quantitative description of short-term plasticity at excitatory synapses in layer 2/3 of rat primary visual cortex, Journal of Neuroscience 17 (20) 7926-40 (1997) [6] Mather, G., Verstraten, F., Anstis, S. (Eds.), The motion aftereffect : a modern perspective, Cambridge, Mass : The MIT Press, 1998.

Figure 1: Activity waves resulting from objects moving at 1 unitl/timestep (bottom), 2 units/timestep (middle), and 3 units/timestep (top).

Figure 1

Figure 2: Activity waves resulting from an extended bar rotating at 10 degrees per timestep.

Figure 2

Figure 3: Translational Motion Aftereffect: Activity waves generated by a static test stimulus after conditioning an excitable membrane with repeated moving stimuli. (a) Conditioning stimulus, timestep = 1; (b) Conditioning Stimulus, timestep = 30; (c) Static test stimuli, timestep = 70; (d) Activity waves resulting from static test stimuli, timestep = 76; (e) Activity waves resulting from static test stimuli, timestep = 82; (f) Activity waves resulting from static test stimuli, timestep = 88.

(a)

(d) Figure 3

(b)

(e)

(c)

(f)

Figure 4: Rotational Motion Aftereffect: Activity waves generated by a static test stimulus after conditioning an excitable membrane with a continuously rotating stimulus. (a) Conditioning stimulus, timestep = 1; (b) Conditioning Stimulus, timestep = 44; (c) Static test stimuli, timestep = 154; (d) Activity waves resulting from static test stimuli, timestep = 160; (e) Activity waves resulting from static test stimuli, timestep = 164; (f) Activity waves resulting from static test stimuli, timestep = 175.

(a)

(d) Figure 4

(b)

(e)

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(f)