Time-Space Transform Model for Timing
The model is composed of three parts:
- Part 1 generates time oscillations over several orders of magnitude. A large number of processors are involved in order to achieve a stable average oscillations activity.
- Part 2 accumulates neuron discharges over space in order to transform time into space locations .
- Part 3 ensures perfomance improvement over time (i.e., learning). A self-organizing map implements the learning of target durations. Timing is achieved by an unsupervised learning paradigm, which means that the performance improves just by sampling the environment.
- Arousal effects modulate the default oscillation frequency provided by the neurons of the time oscillations generator (part 1) . - Attentional effects modulate the efficiency of the connection between the time oscillations generator (part 1) and the time- space accumulator (part 2).
Figure 1 displays the 3 parts of the Time-Space Transform Model for Timing (TSTMT) and the localization of the activation and attention effects. There are no inputs to our model: the oscillations are self-generated.
1/ Time
Two sets of neurons are used in the time oscillations generator. The first set is composed of oscillatory neurons firing at an average frequency of 15 Hz. The second set is based on spiking neurons. These spiking neurons can be clustered in several sub- assemblies that mimic a set of layers, with excitatory connections only between neurons of adjacent layers. Each neuron of a given layer is connected to one and only one neuron of the following layer. There is no intra-layer connectivity. As shown in figure 2, there are 5 layers of one hundred neurons each, with the first layer composed of the oscillatory neurons. All spiking neurons have the same characteristics: a threshold at +100, connection weight at +50, an hyper-polarization value of -30, with an automatic increase until 0 at the rate of +10 per millisecond (ms).
Fig. 2. The time oscillations generator. The first layer of neurons (black) are self-oscillating neurons. The other neurons (gray) are spiking neurons. There are 100 neurons per layer, and only one connection per neuron to an unique neuron in the following
layer.
The average frequencies generated by the time oscillations generator are given on Table 1. The average frequencies decrease as the neuron layers are farther from the oscillatory neurons (layer 1). This decrease is due to the fact that it takes a certain amount of inputs to reach the threshold value, and these inputs are less and less frequent as the neuron is localized farther from layer 1.
Layer 1 2 3 4 5
Frequency (Hz) 15 6 3 0.7 0.1
Layer 1 2 3 4 5
Table 1. Average frequencies generated by the time oscillations generator. The first layer exhibits a spontaneous activity at 15 Hz.
2/ Space
AJOUTER ICI OU + LOIN A QUOI SERVENT LES 3 CARACTERISTIQUES PRINCIPALES
In this module, the spikes coming from the previous module are transformed into a spatial representation. The goal is to have a spatially localized activation depending on the duration. To achieve such results, the accumulator elements - one after the other - are going to count the spikes. In order to have an organized activation of the accumulator module, the number of connections received by each accumulator element are different. The most connected one should reach its activation threshold before the others. It will then reset all the other elements that have not yet reach their activation state. This mechanism (reset) is at the origine of the time-space transform. The accumulator elements are spiking neurons organized in pairs. Each pair of two neurons has the behavior of (=acts as?) a bistable unit, which means that the pair can be considered as either activated or inactivated.
The unit is activated when it has received a certain amount of inputs from the time oscillations generator. In fact, each bistable unit receives the inputs from 10 to 100 neurons of the same oscillation layer. As soon as one bistable unit reaches its threshold, it will send a strong inhibitory signal to all other bistable units of the same layer. All bistable units that are not activated at that time will become hyperpolarized. The effect will be very similar to a reset of a counter. The bistable units that were hyperpolarized will start to count again from 0. In this manner, between two successive activations of bistable units of the same layer, there is a minimum duration which depends directly on the number of connections received from the oscillatory generator. As time passes, the duration interval between two successive activations in the accumulator increases. The units with most connections having been activated earlier, the one that are left are the units with fewer connections.
A bistable unit is made of a pair of neurons as shown on fig. 3. The upper neuron counts the inputs, and may be hyperpolarized from time to time by its neighbors. When this neuron reaches its threshold value and sends a strong inhibitory signal to all others neurons of the same layer, it also sends a strong excitatory signal to its lower part. The lower neuron has a recurrent connection that will keep itself activated once excited. Also, it will continuously send a strong inhibitory signal to the upper neuron, in order to forbid it to enter again into the competition. This way, bistable units change their activation state only once.
Fig. 3. Three bistable units of the time accumulator. The upper neurons (light gray) have a threshold value of 100. They receive the inputs (weight=+3) from the oscillations generator (not shown). They also send inhibitory connections (weight=-40) to all
other upper neurons of the layer. Each upper neuron has an excitatory connection (weight=+100) to its corresponding lower neuron The lower neurons (dark gray) have an inhibitory connection (weight=-50) with their upper neuron, and they also have a
recurrent connection (weight=+100) in order to remain activated once excited.
There are 15 bistable units per layer, much less than the number of neurons in the oscillations generator part. If we place the bistable units of each layer relatively to the number of connections they receive from the oscillations generator, then we can see time as it passes (fig. 4). In this figure, each bistable unit is represented by only its lower neuron, and the bistable units with more numerous connections are on the left.
Fig. 4. In red the bistable units that are activated as time passes. The duration (t) is indicated in millisecond below the corresponding activation of the temporal accumulator. As we can see, as time goes the number of activated units grows. There is a definite amount of regularity in the way the bistable units are activated. This regularity is respected through out repetitions.
The accumulator makes a spatial representation of time.
3/ Learning
A self-organizing map is used for memorizing the accumulator configurations of activation. Each neuron of the self-organizing map (SOM) represents a specific duration. Fig. 5 shows the weight patterns of 5 SOM neurons at initialization and after 500 learning iterations.
Fig. 5. The connection weights between the 5 self-organizing map neurons and the time accumulator. The color scale goes from dark green (-1) to strong red (+1). 0 is represented by a black color. The learning base is composed of a set of 40 accumulator configurations randomly taken between 0 and 3000 milliseconds. One learning iteration uses one example of the learning base to
update the weights of the SOM. 500 iterations means that there is 500 updates of the weights.
The learning rules used to update the weights are the classical equations of the SOM ( [5], pages 44-47). We indicate here the complete learning algorithm we used:
1st step : compute a distance between what the SOM neuron represents and the current time accumulator configuration, using the following equation:
Yj = Sigma |Xi – Wi,j| where Xi is the ith input value, Wi,j is th eweight of the connection between the ith input and the jth neuron, and Yj is the computed distance for the neuron j.
2nd step: Select the neuron number k that displays the minimum distance:
k = arg Min (Yj) ;
3rd step: Update the weights of neuron k using the following equation:
Wik = Wik + alpha * (Xi – Wik)
and update the weights of the 2 neighbor neurons of k (i.e, k-1 and k+1) if they exist:
Wik-1 = Wik-1 + beta * (Xi – Wik-1) Wik+1 = Wik+1 + beta * (Xi – Wik+1)
During the initial deployment of the SOM, alpha=beta=0.1.
When it comes to learn a specific duration, then the number of learning iterations is 5, and the values for alpha are +0.5, and beta=0.
The self-organizing map is famous for its unique ability to respect both the input space topology and the learning base examples distribution. This is the reason why adjacent neurons code for not so different durations and why durations increase as we move along the SOM. By doing 5 learning iterations with the targeted duration, and using bigger values for the alpha parameter, we modify the density probability distribution of the samples, and therefore have a better representation of the desired duration. If we keep the SOM in its learning mode, then the performance in recognition, as also in generation, will improve automatically - just by being submitted to a given duration.
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Simulation results
We run several experiments in order to compare the output of TSTMT with human performance. The first set of experiments is devoted to study the distribution of error when producing a duration. The model obeys the scalar property (fig. 5), which is a first argument for its validity. Moreover, the distribution of produced durations is Gaussian as in typical timing tasks (fig. 6). The second set of experiments demonstrates the effects of attention and arousal (fig. 7). Again, a comparison is made with human performances when manipulating the same effects. Attentional and arousal effects do cumulate themselves. The ability for TSTMT to easily take into account both effects is not shared by other models of timing. The last set of experiments
reports the number of neurons that become activated over time in the accumulator module (fig. 8). These results are to be read at the light offered by the EEG measures of neural build-up activities. Here again, simulation results are similar to human recordings.
Fig. 5. Scalarity of the temporal estimates: distributions superimpose when rescaled relative to the target durations (here targets are 400, 600 and 800 ms, set equal to 1 on the abscissa). Our simulation results (left) are similar to human performance in temporal discrimination: when subjects judge intervals as being equal or not to a pre-memorized target, the
proportion of responses “equal” depends on the current interval-to-target difference, whatever the target value (right:
adapted from Wearden & Bray, 2001). This is a strong argument for the validity of our model.
Fig. 6. This figure represents the Left: distribution of produced durations over 100 training and testing trials with a target duration of 1100 ms. For each trial, a new set of initial weights is randomly generated for the self-organizing map. The arousal
level is set to normal in this simulation (15 Hz). A Gaussian distribution peaking around the target value is obtained (IL FAUT obtenir ça – avec + d’essais?), in accord with typical data issued from temporal production tasks [right - adapted
from Macar & Vidal 2002? A INSERER ].
Fig. 7 Arousal and attentional effects.
The attentional process acts at the level of the link between the oscillations generator module and the accumulator module (see Fig. 1). A reduction of attention is simulated by a diminution of the connection strength, which results in lower accumulation rate, and, hence, longer produced durations. A strength of 80% means that 80% of the spikes generated at
the oscillations generator will reach the accumulator module. In this figure, that averages 20 different training and testing trials, we have set the normal attentional level at 60%. A value of 80% means more than normal attention, as
induced by particularly demanding situations. The arousal effect interferes with the average discharge value of the first neural layer of the oscillations generator. The normal frequency of discharge has been set to 15 Hz. A value of 20 Hz reflects stronger arousal. As we can see, the generated duration is longer with stronger arousal and more attention. Our simulation results are similar to the performance obtained in a task of temporal production with variable attention
and activation levels (right: adapted from Burle & Casini, 2001).
Fig. 8. Number of accumulator neurons recruited as time passes in two conditions of different attentional levels. With an attention of 60% rather than 80%, it takes longer to recruit the same number of neurons. For example, to recruit 38 neurons with a normal attentional level, a duration of 1700 ms versus 1300 ms is necessary with an attentional level
of, respectively, 60 and 80%. Our simulation results are compatible with human performance in temporal production tasks (such as in fig. 7, right panel) and with indices of brain activation showing a positive relationship between the attentional level and temporal estimates (fMRI data: Coull et al., 2004; CNV recordings: Macar et al.,
1999). This figure averages 20 different training and testing trials.
JE NE CROIS PAS qu’on puisse intégrer une fig. CNV ici, et il faudra une explication circonstanciée de cette question ds la discussion, comme on en a déjà parlé. Qu ‘en pensez-vous ?
Claude, les pages couleurs coûtent cher - PEUT-ON remplacer les couleurs par des nuances de gris ds les fig. 2, 4 et 5 ? (et bien sûr 6 et 8, là c’est facile)
ci-dessous des restes éventuellement utiles de la publi de l'ESANN 2005.
One important feature of our model is that it includes few a priori constraints: most properties emerge from the functioning of the entire network population rather than from arbitrarily well tuned properties of individual neurons or inter-neuron connections.
As such, the model is able to reproduce well-established properties of temporal performance as well as data issued from brain imaging studies [10]. First, it accounts for the scalarity observed in humans and animals without introducing any Gaussian-like variance. Secondly, it fits with the monotonous relationship found between brain activation and estimated time in regions thought to subserve the temporal accumulator [8]. In our model, this relationship is paralleled by a lengthening of the estimated time as the number of active neurons in the accumulator module increases. Thirdly, the effects of arousal and attention processes are simulated at distinct levels, as required by behavioural data [3]. While arousal effects can impinge on the frequency generator module, attentional changes affect its link with the accumulator module. By decreasing the number of activated neurons in the latter, our simulation accounts for the finding that decreased levels of attention paid to a target duration correspond to its underestimation (review in [9]), and, furthermore, to a decrease in the activation of brain regions that subtend timing performance [10].
Although certain aspects of our model need further specification, we view it as a first step toward integration of major behavioural and neurophysiological data in sub- and supra-second timing.
References
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[2] J. Gibbon, R.M. Church and W.H. Meck, Scalar timing in memory, In Gibbon, J. & Allan, L. (Eds.), Timing and time perception, New York Academy of Sciences, 423, New York, pages 52-77, 1984.
[3] B. Burle and L. Casini, Dissociation between activation and attention effects in time estimation: Implications for internal clock models.
Journal of Experimental Psychology: Human Perception and Performance, 27, pages 195-205, 2001.
[4] T. Kohonen, Self-Organization and Associative Memory, Second Edition, Springer Series in Information Sciences, Vol. 8, Springer Verlag, Berlin, 1987.
[5] C. Touzet, Les réseaux de neurones artificiels : introduction au connexionnisme, EC2 éd., Paris, 1992 (available online at http://www.up.univ-mrs.fr/Local/umr_6149/umr/
page_perso/Touzet/Les_reseaux_de_neurones_artificiels.pdf).
[6] J.H. Wearden and S. Bray, Scalar timing without reference memory: Episodic temporal generalization and bisection in humans, Quarterly Journal of Experimental Psychology, 54B (4), pages 289-309, 2001.
[7] F. Macar, S. Grondin, and L. Casini, Controlled attention sharing influences time estimation, Memory and Cognition, 22(6), pages 673- 686, 1994.
[8] F. Macar, F. Vidal and L. Casini, The supplementary motor area in motor and sensory timing: evidence from slow brain potential changes, Experimental Brain Research, 125, pages 271–280, 1999.
[9] S.W. Brown, Attentional resources in timing: Interference effects in concurrent temporal and nontemporal working memory tasks, Perception and Psychophysics, 59, pages 1118-1140, 1997.
[10] J.T. Coull, F. Vidal, B. Nazarian and F. Macar, Functional anatomy of the attentional modulation of time estimation. Science, 303, 5663, pages 1506-1508, 2004.
