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Self-organization of neurons in motor cortex for coding directional information
The motor model consists of a
two dimensional heterogeneous self–organizing map with Ne excitatory and
Ni inhibitory spiking neurons.
Each neuron in the output map is
fully connected to the input layer and is linked by probabilistic connections
with other competitive neurons. The sparse connectivity with a Mexican-Hat
profile in the competitive layer plays an important rol in shaping the
network response. The activation function of the neurons is given by the
membrane potential equation 1 in SRM model. The
neural response of an output unit evolves over time by combining the afferent
signals with lateral excitatory and inhibitory feedback.
A detailed description of the model
is given in section 1 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 with the SpikeNNS distribution.
Note. Here we present findings obtained
from the analysis of different network organizations resulted from training
the SOM in different conditions and with different patterns sets (i.e.,
coding 8 or 12 motion directions).
Spiking self-organizing feature maps
In a self-organizing map of spiking
neurons, we can apply learning as a function of the information carried
out in the timing or order of the spikes. Such an approach has been proposed
by Ruf and Schmitt (1998). When an input vector s is applied to
the competitive network, each output node receives a weighted sum of the
afferent signals over all input units. The product w * s represents
a measure of similarity between the two vectors with respect to the Euclidean
distance. Hence, the earlier an output node fires, the more similar its
weight vector is to the input vector. Therefore, the competitive node whose
weights represent the best match of the input vector will fire first and
will represent the winner of the learning step.
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dWij = h * (s - Wij) (Tout - Tj)/Tout (1)where Tj is the firing time of the j neuron, Tout is a time out limit, and h is the learning rate. Instead of the classic topological neighborhood, a temporal neighborhood of the winner is created, so that learning applies only to the neurons which fire until Tout. Adaptation of the weights is also scaled by the distance between the firing time of the competitive neuron and the firing of the winner (last term in the rule).
Our learning algorithm is an adaptation of the learning rule proposed by Ruf and Schmitt (1998). In addition to their model, we have implemented learning for the lateral connections, for both excitatory and inhibitory synapses. Learning is adapted from the rule (1), in such a way that potentiation of synapses predominates over depression. Learning for both afferent and lateral connections is applied as a function of single spike events, only to synapses of the firing units that are placed within a spatial and a temporal neighborhood of the winner. In our algorithm the winner is randomly selected from the subpopulation of units that fire the quickest in one simulation step. For a detailed description of the learning algorithm see chapter 6 in pdf documentation.
Regarding the biological plausibility of the learning algorithm described above, we refer the reader to the studies on visual information processing by Thorpe and colleagues. They suggested that during visual object recognition, the brain does not have time to evaluate more than one spike from each neuron per processing step. Accordingly, Thorpe and Gautrais (1998) proposed a coding scheme based on the time to first spike, where only the first spike of each neuron counts. This idea was supported by other experimental studies, which have showed that most of the information about a new stimulus is conveyed during the first 20 or 50 ms after the onset of the neuronal response.
A self–organizing feature map represents a means of visualizing in a reduced dimensional space (usually two) the spatial relations existing in a multidimensional input space. Hence, a crucial aspect in modeling the development of directional selectivity in motor cortex as a Kohonen self-organization process is to find the training data available for this process. By contrast to the visual or somatosensory feature maps, the nature of the input data for the self-organization of the motor cortex is less obvious.
The milestone of simulation was the formation of a training set, which encodes the directional information needed for learning and moreover, which may yield biological plausibility. Our hypothesis was that input vectors that represent opposite directions are highly dissimilar. Conversely, topologically close to each other directions are encoded by vectors with similar values. Therefore, the input vectors were created to reflect the levels of similarity or dissimilarity between a number of N directions of movement (i.e., N is 12 or 8). See more on the creation of these patterns in chapter 6 in pdf documentation.
With respect to the biological plausibility of the training data, our hypothesis is that the input information can be provided by proprioceptive signals arriving from the muscles involved in a movement (Theeuwen et al., 1996). Recent research on the contribution of muscles to joint torque demonstrated that the activation of bi-articular muscles vary with the direction of force exerted, while mono-articular muscles show significant direction-dependent activation (Bolhuis et al., 1998). Furthermore, it was shown that the mono-articular muscles have preferred movement directions, which cluster over subjects for both force direction and arm posture. This data suggest that the motor unit activity may provide the directional information required for the organization of the cortical directional map. Future work is aimed at training the motor network with experimental data collected from arm muscles while directional movements are performed.
Training of the pulsed neural network
start with neurons being equally responsive to all input patterns (see
Figure
1). With the advance of simulation time, the number of neurons that
respond to a particular input vector slowly decreases, as selectivity of
the neural response
increases. In the end, in the trained
self–organizing map, only a small number of patches of activity occur for
each directional command. These are usually organized around the winners
of the corresponding direction.
Figure 1 shows the response of a trained
network when different patterns are applied. Test directions are located
in a circle section of 120 degrees (i.e., N, NW, W, SW).
(a) Directional pattern 1 |
(b). Directional pattern 41 |
(c) Directional pattern 4 |
(d) Directional pattern 34 |
Figure 1. The trained SOM response
when different patterns are applied. Firing units are shown in red, refractory
period is shown in green and blue units are silent. Note the distributed,
however distinct representation of each pattern.
Winner and lateral neurons
The distributed representation of each directional pattern is a result of the collective organization of both afferent and lateral weights. The organization of afferent weights is reflected in the winner neurons behavior. These are neurons which during training have constantly fired the quickest at the application of an input pattern. Hence they have developed preferences for certain input patterns, based on the selective strengthening of their afferent weights. With respect to their response, about 85% of the winners win for more than one direction of movement. Conversely, we have also found about 15% of winner neurons whose first spike is constantly correlated with only one direction of movement.
The organization of the lateral weights leads to the emergence of a second type of directionally selective neurons, namely the lateral neurons. These neurons have small values on the afferent weights and during testing they never won for any of the directions involved. However, when we analyze in a larger time interval their firing behavior, it occurs that they spike later than the winners and their discharge rates are tuned to the direction of movement. We refer them as lateral neurons, due to the fact that their spiking activity is mainly due to the integration of lateral excitation. Figure 2 shows a self-organizing map labeled with the neurons preferred directions, computed as the movement direction for which the neuron discharge rate is highest. Winner neurons are shown in black, lateral neurons are shown in blue. Different organizations, residing in different numbers of winner and lateral neurons and various types of interactions between them, may be obtained by varying the initialization and training conditions.
Figure 2. Self-organizing map labeled with the neurons preferred directions, computed as the movement direction for which a neuron discharge rate is highest. The level of neural tuning (normalized) is represented by the thick line. In the left side, an assembly of neurons is delimited, composed of winning and lateral neurons whose collective firing encodes the movement direction towards North. The gray arrows indicate the excitatory connections from the neu-rons which fire first (winners) to the neurons which fire later (lateral units).
The joint activation of the winner and lateral neurons give rise to a sort of collaborative cell assembly, which enhances the strength of excitation between neurons tuned to similar directions of movement and suppresses the response in opposite directions. On the left side of Figure 2 the approximate boundaries of the cell assembly coding for movement direction North are indicated. The most important effect of collaborative cells assembly formation is the emergence of a population coding (see bellow).
The collaboration between neurons
leads to the formation of a population code which is supported by the plastic
horizontal feedback system. The lateral connections strengths are not static,
but they evolve together with the afferents. During learning, the lateral
correlations
between neurons fall off with distance
and become stronger only between neighbors with similar directional selectivity.
Our results indicate that neurons that develop similar directional selectivity
become functionally correlated. In the trained map we found that the strength
of the connection between neurons in a pair becomes negatively correlated
with the difference between their preferred directions. This is an important
modeling finding, which is in agreement with experimental estimations of
the weights between motor neurons, as being a function of the difference
between preferred directions (Georgopoulos et al., 1993, Lukashin and Georgopoulos,
1994).
Population coding
With respect to individual cells response
to movement directions, our findings indicate that neurons are broadly
tuned to several directions of movement. Each movement direction is represented
in the map by the activity of an entire population of neurons (i.e., the
collaborative cell assembly). In Figure 3a are shown more than 40 neurons
that participate at the encoding of direction 1 (towards North) by firing
at various rates. Directional information is encoded in the activity of
populations of neurons broadly tuned to similar directions. The population
vectors for 12 directions of movement computed from the individual contribution
of each firing neuron is shown in Figure 3b.
(a) Population coding of movement direction |
(b) Population vectors |
movement. Each vector is a resultant of individual neuron contributions.
Each directionally selective unit
activity is highest for a movement in a particular direction and decreases
with movements further away from that direction. Figure 4 shows the discharge
rates (normalized) of four neurons for different directions of movement.
Neurons have unimodal tuning curves ranging from cells with the width of
the curve of 120 (neuron 3 responds to maximum 5 directions of movement)
to cells with a curve width of 30 (not shown in figure). The median of
the curve width is at 50.
Figure 4. Discharge rates (normalized) of four directional selective neurons plotted as a function of movement directions. The preferred directions of neurons are located at approximately 90 degrees from each other.
Our results are in agreement with the experimental findings of Amirikian and Georgopoulos (2000), which indicate that motor cortical cells are more sharply tuned than previously thought (i.e., do not fit the cosine function). However, our results must be interpreted with care, as different initialization conditions and training procedures of the network can influence the width of the tuning curves. Out model can be used as an exploratory tool for the investigation of the role the neural and connectivity parameters can play in the motor cortex organization.
Neurophysiological implications
Our modeling work on the self organization
of motor cortex represents a first attempt to provide a learning scenario
for how motor cortical neurons develop directional selectivity. Put generally,
it demonstrates that a self-organizing map can learn to distinctively command
different directions of movement, by developing broad neural selectivity
and distributed representations.
Our findings have neurophysiological
implications to a number of hypothesis:
- Emergent vs. innate directional selectivity of motor cortical neurons ?
- It is beyond the scope and resources
of our research to offer an answer to the question of whether directional
selectivity is a genetically encoded feature of the motor cortex or is
acquired by experience. However, this self-organizing motor model proposes
a number of computational hypothesis on what it takes to develop by unsupervised
means, neural selectivity and population coding in a biologically plausible
motor system. By the development of motor neurons' optimal responses, in
contrast with having them pre-wired, the system has a flexible and plastic
architecture that can adapt to the resources available and to the particularities
of the input space.
- Can be the motor cortex modeled like the visual cortex?
- The self-organizing motor model proposed
was inspired by the simulation work on the self-organization of visual
feature maps (see Miikkulainen, Sirosh, and Choe, 1996). It is a common
belief today that development of cortical preferences may be based on a
few design principles, which in turn rely on very general developmental
mechanisms utilizing the input structure of the system (Douglas and Martin,
1991; Niebur and Worgotter, 1993). Our assumption was that developmental
principles best understood for sensory areas represent general laws of
cortical organization and can be applied for cortex modeling. If
our model will be validated by future work, than it provides computational
evidence for the generality of the mechanisms employed. It can also help
to develop new hypothesis on the functional principles of the motor cortex.
One hypothesis considers that the lateral feedback system can play similar
roles in the organization of visual and motor areas. There are a number
of neurophysiological studies which provide evidence on the importance
of motor cortex horizontal connectivity and plasticity (Georgopoulos et
al., 1993; Hess and Donoghue, 1994).
- Computational evidence for the importance of plastic horizontal connectivity in motor cortex self-organization. In our model, plasticity of both excitatory and inhibitory connections is essential for self organization to occur, by finely adjusting cells tuning level to the input space features. One important finding was that the strength of the lateral connection between directionally tuned motor neurons becomes negatively correlated with the difference between their preferred attributes. This computational result supports the hypothesis concerning the manifestation in the brain of a general principle for horizontal connections organization. It is generally believed that this is reflected in the correlation between the strength of interaction and similarity among units’ preferences (Lukashin and Georgopoulos, 1994).
- Is directional information also encoded in the first spike of individual neurons ?
In the self-organizing motor map we found that in the case of about 15% of the winner neurons the first spike is constantly correlated with only one direction of movement. The rest of 85% of the winners win for more than one direction of movement. This finding suggests that directional information may also be read read out from the timing of the first spike of fast responding neurons. Surprisingly, this observation comes out as a possible common feature of information processing in the visual and motor brain.
Recent experiments on visual categorization
revealed the existence of a very fast processing of information in the
visual cortex, possibly based on the order or timing of a single spike
per neuron (Thorpe et al., 1996; Thorpe and Gautrais, 1998). In the case
of the motor system, the influential work of Georgopoulos and his co–workers
(1984, 1986) proposed the population coding scheme as the main paradigm
used to interpret and predict movement based on the motor cells’ discharge
rates. Based on our modeling results, we suggest that a fast response of
the motor cortical areas, read out from the timing of the first spike of
optimally tuned
neurons is certainly advantageous
and quite likely to be implemented by the motor system. However, such an
answer has a limited precision and only further processing of the directional
information by a large population of cells can give rise to an accurate
movement. For a more detailed discussion of our results we refer the reader
to chapter
6 and chapter
7 in pdf documentation.
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