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 Motor model
 Visuomotor model

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• Visual coding of movement direction
• Learning algorithm
• Results
• Visuomotor, signal-related and movement-related neurons
• Neurophysiological implications

The visuomotor model consists of two networks (4 layers), each network with an input and hidden layer and specific patterns of connectivity. The motor network is a self-organizing map, with the input layer fully connected to the hidden layer and a sparse probabilistic connectivity within the competitive layer. The motor SOM used in the visuomotor model is a self-organizing map trained to represent a set of directional patterns as described here. The visual network consists of a layer of directionally selective neurons connected through receptive fields with the visual input layer. Each hidden neuron in the network receives afferent connections from one input layer and lateral connections from other neurons in the same area. The visual cortical layer is fully connected to the motor map. Full connectivity is required at this level to ensure that any population of motor neurons has access to all visual directions of movement. 

A detailed description of the model is given in section 2 chapter 6 in pdf documentation. In SpikeNNS html manual one can found instructions on how to build, initialize and train the model. The network implementation, together with the pattern train and test files, configuration parameters and a trained network are provided within the SpikeNNS distribution.. 

Visual coding of movement direction

The visual network consists of a hidden layer of directionally selective neurons that have receptive field from a retina-like layer. The visual map has a built-in, basic capacity of signaling motion direction. It is today well established that the cortical analysis of visual space relies on the functioning of a fundamental neural machinery, referred to as a hypercolumn. A hypercolumn represents a set of columns which are responsive to lines of all orientations from a particular region in space via both eyes, and to movements in directions orthogonal to the orientation axes (Kandel et al., 2000). In our model, the scale of a hypercolumn dimension and the range of lateral connectivity represent a major simplification. One hypercolumn consists of 4x4 neurons, each firing for only one preferred direction and being silent for movements in different directions. Excitatory lateral synapses connect each neuron with the first order neighbors that have the same preferred direction. 

What we actually simulate in the visual network is the following process: the retina layer is presented with a bar moving in a certain direction. Than, the hidden visual cells that have retinal receptive fields along the trajectory of the moving stimulus and whose preferred directions are equal with the stimulus motion direction become activated. A demonstration of how this happens in a model of a visual orientation map is given by Bednar and Miikkulainen here. We believe that a developmental model of a visual directional map can be integrated in our architecture to offer it more biological plausibility, but without changing its functionality as it was abstracted by us in our current model.

Learning procedure

Prior modeling work of visuomotor mapping using correlation-activity associative learning was based on rate coding neurons, where learning is applied as a function of neural discharge rates (Salinas and Abbott, 1995; Baraduc et al., 1999). The contribution of our model to the modeling of visuomotor modeling by unsupervised means has two aspects: 

• is based on computations with spiking neurons that can naturally perform detection of temporal correlation 
• the visuomotor mapping is learned in the conditions of a biologically realistic motor population coding of motion direction.

1. the input motor units generate a command for movement in a random direction d. This input elicits in the motor map the activation of the neural assembly involved in the representation of the direction d. The hidden units activity is maintained for about 400 ms.
2. after a certain delay, required for the movement to be visually perceived, the retina is presented with a bar moving in direction d. This motion persists until the end of the learning cycle. After another delay, the retinal information reaches the visual network, where the motion selective cells fire and signal the direction. 
3. the visuomotor connections transmit the neural activity from the visual to the motor area with an average delay of 60 ms. 
4. during a time window of approximately 200 ms while both networks are active, spike-timing dependent learning is applied to the visuomotor interlayers connections. The connection between a visual presynaptic neuron and a motor postsynaptic unit is strengthen as a function of the interval x that separates the time of the postsynaptic potential peak (e.g., equal to the presynaptic spike time plus the spike transmission delay) and the postsynaptic spike moment: 

 
w (t+1) = w (t) + h * [ sum(e) - w(t)],      with e =  exp(-x/Mc),  if x > 0 and e = 0, if x < 0


where the sum is computed over all presynaptic spikes emitted on the synapse w, Mc is the membrane time constant and h is the learning rate. The weight of the connection has an upper limit equal to the total effect of the presynaptic spikes on the postsynaptic unit.

To test the learning of visuomotor mapping of direction, the retina is presented with a bar moving in a constant direction for a certain time interval, while the activity evoked in the motor map is recorded. For each direction of motion, the motor activity is analyzed with respect to the shape of the firing patterns elicited (i.e., neurons involved) and to the population vector resulting. Figure 1a shows the discharge rates of the motor neurons activated by the visual information coding the movement in direction 1 (North). Note that the firing patterns elicited in the motor network in the visual condition (i.e., when the motor map is exclusively driven by the visual stimuli) and in the motor condition (i.e., when a movement command is issued) (Figure 3a in motor model) are very similar.
 

(a) Motor population in visual condition

(b) Motor population vectors in visual condition

Figure 1. (a) Population activity occurring in the motor network in the visual condition, while a movement in direction 1 (North) is perceived. The network activity elicited by the visual input resembles very closely the motor population representation of movement in the same direction (see Figure3a ). (b) The motor population vectors computed during the visual condition for several movement directions.

As concerns the organization of the inter-cortical weights we found a negative correlation of the connection strength with the difference between the preferred directions of visual and motor neurons in a pair. The peak of the inter-layer synaptic strength corresponds to a difference between PDs of 30 degrees (i.e., compared with 0 for the motor map). This result suggests a broader coupling of neurons, which has an effect on the accuracy of the generation of the desired direction of movement.

The role of inter-cortical connections delay in visuomotor mapping accuracy

The accuracy of  visuomotor mapping is a function of two factors: 

• the accuracy and distinctiveness of directional patterns representations in the motor network 
• the values of the noisy delays associated with the inter-layers spike transmission.

(a) Directional pattern 1 for D1 = 59 ms.	(b) Directional pattern 41 for D1 = 59 ms.
(c) Directional pattern 4 for D1 = 59 ms.	(d) Directional pattern 34 for D1 = 59 ms.

Figure 2. Visuomotor model in action. In the left part of the network, observe the neural activity evoked in the motor map when the visual network is activated through retina input. Activities elicited by four different patterns are shown. See how the motor representation shifts from pattern 1 to pattern 34. Note the differences to the motor representation in the motor condition here. 

The motor network activity evoked in the visual condition overlaps the network response in the motor condition (see Figure 1) however, they are not perfect matches. More interestingly, the number and the nature of neurons which are activated in the motor network by the visual input is a function of the delay values of the inter-layer connections. 

In our learning algorithm, for a connection between two neurons to be strengthened, the presynaptic visual spike which travels along the axon for about 60 ms (i.e., time interval whilst the postsynaptic neuron can fire up to 8 times) has to reach the postsynaptic motor neuron in a short time window before this one fires. Obviously, this large delay makes the synchronization of the presynaptic and postsynaptic activities more difficult. In general, this large delay should favor the formation of functional connections from visual neurons to those motor neurons that fire at high discharge rates during the whole movement, rather than neurons which fire precisely timed, but fewer spikes. We have discussed the implications of this aspect in chapter 6 in pdf documentation.

Moreover, the strengthening of inter-layer connections strongly depends on the absolute value of the transmission delays. For instance, the networks shown in Figure 2 are obtained for a baseline length value of 59 ms, which occurred to be the value that favored most the strengthening of weights and gave rise to the most accurate visuomotor mapping. Figure 3 shows the motor network response (in 2D visualization) when different baseline values of the delays are used during training. All firing units are shown in yellow. 
 

(a) Directional pattern 1 with D2 = 57ms.	(b) Directional pattern 4 with D2 = 57 ms.
(c) Directional pattern 1 with D3 = 62 ms.	(d) Directional pattern 4 with D3 = 62 ms.

Figure 3. Motor network response in the visual condition for different values of the baseline length of inter-layer connections. The motor network is shown (in left) with a fragment of the visual network (in right). Note that for D2 = 57 ms, the network response is broader than for D1 = 59 ms (in Figure 2) which is also broader than the activity evoked for D3 = 62 ms.

In our model, the length of all connections between the visual and motor layer are initialized with values generated from a baseline value D. Hence, the transmission of all spikes will be affected by noisy delays generated from a gaussian distribution with the same mean. This is clearly a simplification. In a large-scale network model as well as in the living brain, inter-cortical connections which link neurons from different areas introduce different delays in spikes transmission. One can see the length values D1, D2, D3 as associated with different subsets of connections that run between distant areas in the brain. This setting would lead to the formation in the projection area (i.e., in our case motor area) of areas of neurons that exhibit different types of behavior, similar to our network different responses. For instance, connections with D1 and D2 length values will favor the formation in the projection area of broad visuomotor representations. Those with D3 length will determine less neurons to become visually responding and favor motor neurons to remain exclusively responsive to the motor and proprioceptive signals. The delays of the inter-layer connections are one of the factors, besides others, that determine the emergence in the motor network of three types of neural behaviors.

Visuomotor, signal-related and movement-related neurons

As we mentioned above, an interesting result of our simulations is the emergence in the motor network of different neural behaviors. The factors that determine neurons to develop preferences for both motor (i.e., proprioceptive) and visual input or to only one of these inputs (remember that all neurons started equally selective) are discussed in detail in chapters 6 and 7 in pdf documentation. For instance, the neural parameters, such as threshold value, number of input excitatory and inhibitory connections, have an indirect influence, by contributing to the neural response variability which in turn affects the computation mode of the neuron and its spiking behavior. The pattern of connectivity is also important in determining if a neuron becomes a winner for a certain pattern, has a directionally tuned discharge rate, or is not directionally selective at all. The factors that influence directly the dynamics of motor neurons during the visual condition are the inter-layer delay values and the stability and accuracy of the population codes of movement directions. 

All these factors lead to the formation of three distinct neural behaviors:

• visuomotor neurons - show directional selectivity during both movement execution and under visual stimulation. From 264 motor neurons, about 60% become visuomotor responsive. 
• signal- or visually-related neurons - represent about 10% of all units and are characterized by a directionally tuned response only during the visual condition, while they are not directionally selective during movement execution. This category also includes motor neurons that change their directional motor preference in the visual condition. That is, for the execution of certain movement direction they fire only accidentally, no selectivity is shown. However, in the visual condition they become significantly tuned to that movement direction. Hence, their is a shift in their PD between the motor and visual condition.
• movement-related neurons - about 8% of the motor neurons directionally selective during movement control are not visually selective and fire only accidentally in the visual condition.

(a) Directional pattern 3 in motor condition.	(b) Visuomotor neurons for pattern 3 in visual condition with D = 59 ms.
(c) Directional pattern 3 in visual condition with D = 57ms.	(d) Signal-related neurons for pattern 3.
(e) Directional pattern 3 in visual condition with D = 62 ms.	(f) Movement-related neurons for pattern 3.

Figure 4. Illustration of the visuomotor, signal-related and movement-related sets of neurons. In left the motor network response for the directional pattern 3 is shown in different conditions: (a) in the motor condition; (c) in visual condition with D = 57 ms; (e) in visual condition with D = 62 ms. In right the subsets of motor neurons are shown: (b) visuomotor neurons obtained through the intersection of the sets of neurons firing in the motor and the visual condition; (d) signal-related neurons computed as the disjunction between the neurons firing in the visual condition and those firing in motor condition; (f) movement related neurons labeled as the neurons which fire in the motor condition and do not fire in the visual condition.

Note also that neurons change (or rotate) their preferred directions between the motor and the visual condition. For instance neurons with a PD of 210 degrees in the motor condition (i.e., corresponding to pattern 4), rotate their PDs in the visual condition with 30 degrees by becoming tuned to direction 3 (in lower part of figure 4c). 

Our results are in agreement with studies of visuomotor processing involved in reaching that demonstrated the existence of various types of neural activity. During an instructed delay task followed by a pursuit tracking task, Johnson and colleagues (1999) have analyzed the directional discharge of neurons in monkey’s premotor and primary motor cortex. From 240 neurons, in 132 cases, significant directional tuning was found for both the cue and track periods. In 26 neurons, directional tuning was found only during the cue period, and in another 54 the directional tuning was significant only in the track period. They classified neurons in: (1) visuomotor neurons, whose activity show the co–existence of visual and movement control signals; (2) signal neurons, defined as motor neurons with visual properties, which respond transiently to the onset of the visual cue; (3) movement-related neurons that fire only for movement control.

Burnod and colleagues (1999) have also introduced in their model four types of neural behaviors: (1) sensory units that are timed–locked to sensory signals in all domains; (2) motor units, time–locked to motor events; (3) matching units, which learn correlation between sensory and motor signals; and (4) condition units, which store correlation between sensorimotor signals and reinforcement contingencies. In contrast to their model, where different types of behaviors are a built-in feature of the network, our model implements a developmental scenario by which various neural response naturally emerge from the way neurons interact and collaborate with each other.

Neurophysiological implications

What distinguishes our model from previous work in unsupervised learning of visuomotor mapping, is that our architecture is implemented with spiking neurons and learning is applied in terms of single spike events in the conditions of a motor population coding of motion direction. Probably the most interesting results coming out from out model are those concerning the emergence during visuomotor learning of different neural behaviors in the motor network (see above). 
These findings face us with two research questions. First question concerns the role of the spiking neurons within this model. Second question tries to establish a connections between our results and the mirror neurons behavior:

• What brings new the spiking neurons for visuomotor mapping learning ?

Previous models of visuomotor mapping have been successful in obtaining the alignment of visuomotor information, and even more, in learning visuomotor coordinates transformation. Than why use spiking neurons ? Are there any computational benefits from using spike timing codes rather than using the rate code solely ? Our simulations, concerning the motor cortex organization and visuomotor mapping, suggest that both rate coding and temporal coding (i.e., the time to first spike) are used for coding directional information. On the one hand, for the fast control of movements, directional information might be read out, if necessary, from the timing of the first spike of a subset of a fast responding neurons. On the other hand, learning of visuomotor mapping is favored 
by a rate coding of directional information in the motor networks, because it supports the correlation of activity in the two networks.

However, the emergence of different neural responses (i.e., winners vs lateral neurons; visuomotor vs. movement- or signal-related neurons) has become apparent only in the presence of delay coding of information, noisy delays and spike-timing dependent learning. For instance, a motor neuron becomes visually selective only if it fires a spike in a short time window after the postsynaptic potential arrival. Furthermore, the weight of a visuomotor connection has an upper bound limit given by the total postsynaptic potential effect on the postsynaptic spikes. If the contribution of presynaptic spikes at the postsynaptic firing is not significant the weight remains small, and the neuron does not become directionally tuned in the visual condition. If the motor neuron spike synchronizes with the postsynaptic spike arrival, than the visuomotor connection weight increases. Even more, it is sufficient that the visual excitation is received at the moment when a peak of the motor neuron activity is registered, without the motor neuron to fire. In this case only by integrating the visual and motor excitation can the motor neuron fire (i.e., the case of signal-related neurons).
 

• From multimodal neurons to mirror neurons ?

The mirror neural system in the monkey’s premotor cortex (Rizzolatti et al., 1999) has been recently  proposed as the system responsible for the linkage of self–generated and observed actions. The interpretation is that mirror neurons can allow the matching of the neural command for an action with the neural code for the recognition of the same action executed by another individual. Mirror neurons have been observed for reaching and grasping actions, and they are highly specialized for certain types of movements.

However, even if mirror neurons functionality has been recently incorporated in several imitation modeling proposals (Billard and Mataric, 2001; Maistros and Hayes, 2002; Metta and Fitzpatrick, 2002), few attempts have been made, so far, to understand the way they develop such a highly specialized matching property (Arbib, 2001). We are interested to explore in future work if we can bridge the gap between the multimodal neurons obtained in our model and mirror neurons behavior. For doing this we intend to create a large-scale model suitable for learning sequences of arm directional movements. Our belief is that exploring the learning of visually guided sequences of actions with spike processing networks and movement population coding can provide insights into the way the mirror neurons behavior emerges in the brain.