# A similarity and fuzzy logic-based approach to cerebral categorisation

Julien Erny, Josette Pastor, Henri Prade

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Julien Erny, Josette Pastor, Henri Prade. A similarity and fuzzy logic-based approach to cerebral categorisation. European Conference on Artificial Intelligence 2006, Aug 2006, Riva del Garda, Italy. pp.21-25. hal-03361588

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<figure><img src="image_1.png" /></figure>
# A Similarity and Fuzzy Logic-Based Approach to Cerebral Categorisation

Julien Erny<fnref n="1" />,<fnref n="2" />, Josette Pastor<fnref n="1" /> and Henri Prade<fnref n="2" />

**Abstract.** This work proposes a formal modelling of categorisation processes attempting at simulating the way information is categorised by neural populations in the human brain. The formalism mainly relies on a similarity-based approach to categorisation. It involves weighted rules that use inference and fusion techniques borrowed from fuzzy logic. The approach is illustrated by a simulation of the McGurck effect where the combination of contradictory auditory and visual stimuli creates an auditory perceptive illusion.

## 1 INTRODUCTION

The understanding and the prediction of the clinical outcomes of focal or degenerative cerebral lesions, as well as the assessment of rehabilitation procedures, necessitates knowing the cerebral substratum of cognitive or sensorimotor functions. This is addressed by functional neuroimaging studies. They have shown that sensorimotor or cognitive functions are the offspring of the activity of large-scale networks of anatomically connected cerebral regions [10]. However, no one-to-one correspondence between activated networks and functions can be found [11].

To understand the intricate relations between these networks and the cognitive functions, we need a model dealing explicitly with symbolic information while being strongly rooted in our knowledge of the neurobiological processes. Two complementary approaches address the problem. On the one hand, connexionist models focus on neural representation, at different scales. Biologically-plausible modelling deals with individual neurons, achieving a good level of accuracy but losing the symbolic aspect of information [26, 27]. On a coarser scale, simpler neuron models have revealed links between the neuronal substratum and simple cognitive functions [4, 13, 16, 17]. On the other hand, models from symbolic AI tend to reproduce cognitive processes without being interested in biological plausibility [3, 12, 20, 21, 24, 25]. However none of these approaches seems to answer the question of how the activation of large-scale networks follows the brain's information processing. This paper proposes an intermediary layer of interpretation, which echoes some other works [4, 6, 14]. However these works may lack a certain biological plausibility especially in the representation of categorisation processes, problem that will be addressed hereafter.

In the present paper, we first discuss the previous models and describe the requirements that a new model should address. Then we present a new approach that can fulfill some of these prerequisites. We then apply this model to phonemic discrimination. We conclude

with a discussion of the results and some perspectives.

## 2 EXISTING MODELLING APPROACHES

Representing information processing in large-scale cerebral networks necessitates departing from traditional connectionist methods, which tend to represent neural populations as neurons [2]. At this level, symbolic information processing is specific to the neuronal population that implements it and can be assumed to emerge from the numerical processing in the population's individual neurons. The causal connectivity approach [22] integrates this symbolic processing with more numerical aspects linked to the activation levels of cerebral regions in large-scale networks. This approach inspired probabilistic models [14] and has influenced our own work. In causal connectivity, the neuronal populations are seen as information processors while the axon bundles linking these populations are information transmitters. The information itself is represented as two-dimensional data: i) a numeric component, called magnitude, stands for the overall activation of the neuronal population that processed this piece of information (thus allowing comparisons with neuroimaging data), and ii) a symbolic component, called type, that qualifies the pattern of the firing neurons in the population. Information is then spread in the network of populations using a dynamic Bayesian network formalism. The treatment taking place in a node can be seen as categorisation since the type of incoming information is compared to archetypes stored in the node. These archetypes represent the configuration of the synaptic weights inside the population and are modified dynamically every time information passes through, thus implementing an unsupervised, on-line learning mechanism.

However this model is lacking a primordial feature of cerebral categorisation: similarity [23]. It is known that close-by neurons will activate and fire for similar stimuli, suggesting a similarity code between stimuli based on the proximity between neurons responding to them. In the model mentioned above [14], there is no such a similarity and the archetypes are disconnected from each other. As a consequence, the model cannot interpolate between archetypes, losing the ability to compute a fine-tuned response to an ambiguous signal.

The information flowing through the brain is noisy and often uncertain. This is a major issue that has to be dealt with. Moreover, the proposed framework should have good potentials for learning on an unsupervised and online basis. Amidst classical categorisation systems in AI [19], none meets exactly these requirements. However, case-based reasoning [1, 8, 9], and fuzzy rules-based reasoning [9] provide ideas that can be a starting point for a specific formalism.

Our goal is to build a model dealing explicitly with cerebral information, while being as close as possible to neurophysiological reality, and flexible enough to incorporate new knowledge coming from

<footnote n="1">¹ U455, Inserm/Université Paul Sabatier, Pavillon riser, CHU Purpan, 31059 cedex 3, Toulouse, France; {julien.erny, josette.pastor} @toulouse.inserm.fr</footnote>

<footnote n="2">² IRIT, CNRS/Université Paul Sabatier/INPT, 118 route de Narbonne, 31062 cedex 9, Toulouse, France; henri.prade@irit.fr</footnote>
neurosciences. However, learning will not be addressed in this paper.

# 3 A MODEL OF A NODE IN A LARGE-SCALE CEREBRAL NETWORK

A node implements the information processing and propagation by a functionally coherent neuronal population. These mechanisms are based on a functional decomposition of this population.

## 3.1 Principle

### 3.1.1 Decomposition of a node

A node receives information through several inputs and emits one output. For each input, we consider a receiving subpopulation and, symmetrically, we consider another emitting subpopulation. Inside each subpopulation we consider patterns representing, i) in the case of the receiving populations, the neurons responding to a specific configuration of the input, and ii) in the case of the emitting population, the neurons firing to propagate the output. The output domain is discrete since the internal regions of the brain does not handle continuity (every continuous incoming signal, like audio waves for example, is “broken” into categories by primary sensory cortex). Receiving and emitting patterns are linked through inference rules using the following principle : an emitting pattern is linked to one receiving pattern in each receiving population. Hence, there is one inference rule per output category (i.e. emitting pattern). Although a given receiving pattern may be used in different rules, its contribution to the activation of the emitting pattern is rule-dependent. Therefore, what is referred to in the rule is not the receiving pattern *per se* (i.e. the neuron's bodies) but a functional image of this pattern as it is used in the rule (i.e. the activated neurons : bodies and axons). From now on, “receiving pattern” will be used to designate those functional images (see Fig.1).

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Physical view: the neuronal population is partitioned in two receiving subpopulations and an emitting one. Each receiving population contains two receiving patterns connected to the emitting patterns (some of the connections are not shown for readability's sake). The functional view shows how two functional images of the same pattern can be used in two different rules.</figcaption></figure>

### 3.1.2 Patterns

The patterns are formally fuzzy sets. In the case of emitting patterns, there exists one fuzzy set corresponding to each element of the output domain, that element being the core of the set. This element is called referent and characterizes the emitting pattern. The other members of the set will be referents of other emitting patterns and they belong to the set with a degree that reflects the similarity between these other patterns and the considered pattern. As for the receiving patterns, their cores contain the elements of the corresponding input domain that are perfectly recognised while the other members are the close-by and less recognised elements. The next section gives a formalism for these notions and presents the key equations describing it.

### 3.1.3 Inference Principle

Similarity of incoming information is computed against the receiving patterns. The more they are compatible, the more these patterns are activated. When the activity of a pattern exceeds a threshold, the pattern fires. When all the receiving patterns of an emitting pattern fire, this pattern is activated. Taking into account the similarity of the activated emitting patterns, a decision policy is then applied to decide which pattern fires the final output (see Fig. 2).

<figure><img src="image_3.png" /><figcaption>**Figure 2:** Inference principle</figcaption></figure>

## 3.2 Formalism

Let $X$ be a node with $n$ inputs. Information is a couple $(M, T)$ where $M$ is the magnitude and $T$, the type. $M \in \mathbb{R}^+$ and $T$ is a fuzzy set defined on $\mathbb{D}_T$, a discrete set. Let $(M_i^{in}, T_i^{in})$ be the incoming information on input $i$ and $(M_i^{out}, T_i^{out})$ the output information. To keep the notations manageable, we call $\mathbb{D}_i$ and $\mathbb{D}_{out}$ the domains of $T_i^{in}$ and $T_i^{out}$.

Let $\mathcal{BR} = \{R_i\}_{i \in [1,p]}$ be the inference rule base. We call $PR_i^j$ the fuzzy set standing for the receiving pattern in input $i$ for the rule $R_j$, and $PE_j$ the fuzzy set standing for the emitting pattern corresponding to the conclusion of $R_j$. We call $na_i^j$ (resp. $sa_i^j$) the level of activation (resp. the threshold of activation) of the receiving pattern $PR_i^j$. We call $dec_i^j$ the firing level, i.e $\max(0, na_i^j - sa_i^j)$. Finally, we call $dec^j$ the level of activation of the emitting pattern $PE^j$.

### 3.2.1 Activation during reception

For each time step, for each rule $R_j$ and for each input $i$, the compatibility $\alpha_i^j$ of $T_i^{in}$ with $PR_i^j$ is computed as $\max_{C \in \mathbb{D}_i} \min(T_i^{in}(C), PR_i^j(C))$, which estimates the consistency
of the two fuzzy sets. Hence the activation level at time t is computed from the previous activation level by increasing it when $\alpha_i^j$ is sufficiently large and the pattern has not been already fired recently:

$$na_i^j(t) = relax_X^{(1)} \cdot na_i^j(t-1) + k \cdot f_{\text{act}}(\alpha_i^j, \bar{M}_i(t)) \cdot f_{\text{refract}}(dec_i^j(t-1)) \quad (1)$$

where,

* $f_{\text{act}}$ is increasing from $[0, 1] \times [0, 1]$ on $[0, 1]$, and prevents information with a too low magnitude from passing through. $\bar{M}$ is a normalized magnitude ($\in [0, 1]$). $f_{\text{act}}$ should be a threshold function, like a sigmoid for instance.

* $f_{\text{refract}}$ is decreasing from $[0, 1]$ on $[0, 1]$, with $f_{\text{refract}}(0) = 1$ et $f_{\text{refract}}(1) = 0$. It stands for the refractory period of the population, preventing it to activate again when it has been activated strongly before. A linear function ($f_{\text{refract}}(dec) = 1 - dec$) or again a sigmoid can be used.

* $relax_X^{(1)} \in [0, 1]$ is called the relaxation parameter and characterises the temporal integration of the pattern. If it equals 0, there is no temporal integration.

### 3.2.2 Activation during emission

We have to update $dec^j$ for each rule $R_j$. We assume that the combination of the emitting patterns obtained at the previous step is linear with respect to each component, hence,

$$dec^j(t) = relax_X^{(2)} \cdot dec^j(t-1) + \sum_{i=1}^{n} p_i \cdot dec_i^j(t-1) \quad (2)$$

where the $p_i$ are parameters that have to be tuned in order to produce the more accurate simulation, and where $relax_X^{(2)}$ is the relaxation parameter for the emitting subpopulation, defined as above.

When the $dec^j$ are known, we still have to combine and decide which pattern will fire. If there is only one emitting pattern activated ($dec^k \ge 0$), then the task is done and $T^{\text{out}} = PE^k$. However, when several rules $R_{k_1}, \dots, R_{k_q}$ are triggered, there is a conflict that has to be solved. But first we will cumulate the activations, meaning that every time a pattern is mentioned as close to another, they will share their activations. If rule $C$ is the rule from which $C$ is the referent, $dec^{\text{rule}(C)}$ is the total activation of the corresponding emitting pattern (i.e. direct activation summed with similarity activations):

$$dec^{\text{rule}(C)}(t) = \sum_{i \in \{k_1, \dots, k_q\}} PE^i(C) \cdot dec^i(t) \quad (3)$$

When all the activation levels are computed, we just choose the most activated pattern as the firing one.

### 3.2.3 Output magnitude

The magnitude representation is handled as causal connectivity as in [14], by a Bayesian-like propagation mechanism. The output magnitude depends on the one hand on the nature of the node (its function) and on the other hand on the level of compatibility of the incoming information. Formally,

$$M^{\text{out}}(t) = f_X^{(1)}(dec_{\text{max}}(t)) \cdot f_X^{(2)}(\overline{M^{\text{in}}(t-1)}, u) \quad (4)$$

where $\overline{M^{\text{in}}(t-1)}$ is the average over all the incoming magnitudes at $t-1$; $u$ is a random variable standing for the noise, and $dec_{\text{max}}$ is the activation of the firing emitting pattern. The function $f^{(1)}$ is increasing, defined from $[0, 1]$ on $[0, 1]$. Since its purpose is to generate a low output magnitude for something badly recognised, a threshold function would be a good choice. $f^{(2)}$ is a real function that may be non-linear.

### 3.2.4 Example

Let us consider a very simple example to make these equations somehow clearer. The focus will be on the inference mechanism letting aside the temporal aspect. To do so, we will describe the behaviour over one time step $t_0$, in a completely relaxed node (there are no activation carried over from previous steps) and with all activation thresholds reduced to 0 (i.e $\forall i, j, na_i^j(t_0-1) = dec_i^j(t_0-1) = 0$).

Consider a node with two attributes 1 and 2, with domains $D_1$ and $D_2$, receiving respectively inputs $T_1^{\text{in}}$ and $T_2^{\text{in}}$, with weights $p_1$ and $p_2$. The magnitudes for each input are set to 1 and the parameter $k$ is also set to 1. There are three “if...then” rules in the base :

$$(PR_1^1, p_1) \otimes (PR_2^1, p_2) \rightarrow PE^1 \quad (5)$$

$$(PR_1^2, p_1) \otimes (PR_2^2, p_2) \rightarrow PE^2 \quad (6)$$

$$(PR_1^3, p_1) \otimes (PR_2^3, p_2) \rightarrow PE^3 \quad (7)$$

Where $\otimes$ stands for conjunction. It then works in three steps :

* First, for each receiving pattern $PR_i^j$, reception activations $na_i^j$ are computed. In this simplified case, it is just a matching rate between the pattern and the input (e.g. [9]), $t_i^{\text{in}}$ being the input at $t_0$:

$$na_i^j(t_0) = \max_{C \in D_i} \min(T_i^{\text{in}}(C), PR_i^j(C)) \quad (8)$$

* Then, the direct activation $dec^j$ of the emitting pattern $PE^j$ is computed :

$$dec^j(t_0) = \sum_{i \in \{1, 2\}} p_i \cdot na_i^j(t_0) \quad (9)$$

* Finally, the direct activations of the emitting patterns are combined using similarity and according to equation (3). Fuzzy sets $PE^1, PE^2$ and $PE^3$ are shown in Fig.3. In the figure, $c_1$ belonging to $PE^1$ (whose referent is $c_2$) at a level $h_1$ means that the similarity between $c_1$ and $c_2$ is $h_1$. It also means that the activation of the pattern whose referent is $c_1$ will contribute to the activation of $PE^1$ with $h_1$ as a discounting factor. The equation (3) gives us: $dec^{\text{rule}(c_1)}(t_0) = dec^2 + h_2 \cdot dec^1$; $dec^{\text{rule}(c_2)}(t_0) = dec^1 + h_2 \cdot dec^2 + h_1 \cdot dec^3$; $dec^{\text{rule}(c_3)}(t_0) = dec^3 + h_1 \cdot dec^1$, where $dec^{\text{rule}(c_i)}$ is the total activation of the pattern of referent $c_i$. The most activated pattern (i.e. for which $dec^{\text{rule}(c_i)}$ is maximum) is then chosen as the final output.

# 4 APPLICATION TO PHONEMIC DISCRIMINATION

As an illustration we have built with a unique node a very simple phonemic discriminator. This has permitted to reproduce an auditory illusion, the McGurck Effect.

## 4.1 Model

To define a phoneme, we use the articulatory theory of language that characterises a phoneme by the way we produce it. Table 1 represents
<figure><img src="image_4.png" /><figcaption>Figure 3: Emitting patterns</figcaption></figure>

a simplified French phonologic system. The two described attributes (locus and mode of articulation) are our two inputs. The similarity follows what lines Table 1 suggest and we only focus on [b], [d] and [g] (i.e. [b] "is close" to [d] and "less close" to [g] and [d] "is close" to [g]).

<table><caption>Table 1: Consonants in the French phonologic system</caption><thead><tr><th>locus ⇒ mode ⇓</th><th>labial</th><th>dental</th><th>velar</th></tr></thead><tbody><tr><td>plosive voiceless</td><td>p</td><td>t</td><td>k</td></tr><tr><td>plosive</td><td>b</td><td>d</td><td>g</td></tr><tr><td>nasal</td><td>m</td><td>n</td><td></td></tr></tbody></table>

The node is built with already learned categories. Assume there are four receiving patterns: $PR_{bil} = \{(bil, 1.0), (lab, 0.6)\}$, $PR_{dea} = \{(lab, 0.6), (dea, 1.0), (alv, 0.6)\}$, $PR_{vel} = \{(pal, 0.6), (vel, 1.0)\}$ and $PR_{plo} = \{(plo, 1.0), (plv, 0.6)\}$. There are three categories (i.e. emitting patterns) already implemented in the node: [b], [d], [g] and the corresponding rules are shown in expression (10). The parameters are: $k = 0.6$, and $relax^{(1)} = 0.98$ (for eq.1), and $relax^{(2)} = 0.15$ (for eq.2).

$$R_b : (PR_{bil}, 0.5) \otimes (PR_{plo}, 0.5) \rightarrow \{(b, 1.0), (d, 0.6)\} \quad (10)$$

$$R_d : (PR_{dea}, 0.5) \otimes (PR_{plo}, 0.5) \rightarrow \{(b, 0.6), (d, 1.0), (g, 0.6)\}$$

$$R_g : (PR_{vel}, 0.5) \otimes (PR_{plo}, 0.5) \rightarrow \{(d, 0.6), (g, 1.0)\}$$

Figure 4 shows the response of the node to a stimulus [b]. The stimulus is presented during 50 ms, every second, to allow neuronal processing to take place.

We see clearly that only the pattern [b] fires, the other two staying unactive all the time. Now what happens when we change the threshold?

## 4.2 McGurck Effect

Figure 5 shows the principle of this illusion discovered in 1976 by McGurck [18]. The visual [ga] is a person articulating a [ga] while the sound played is a [ba]. Using some recent insights about its neurophysiological nature, we implemented it using the same node but in a modulated way, i.e where the thresholds triggering the category [g] are lowered when a visual [g] is presented along with an auditory [b]. Thanks to the combination of activations in the emitting patterns, the response comes as a [d] (see Fig.6). Because of the lowering of the activation threshold, the emitting pattern [g] gets activated, and the pattern [b] is activated by the stimulus itself. If we look at Table 1, we see that between these two sounds, stands the [d]. Hence, the pattern [d] is close to the other two and gets activated when they are,

<figure><img src="image_5.png" /><figcaption>Figure 4: Activations of the emitting patterns, in response to an auditory stimulation of type [b]</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: The simultaneous presentation of a 'visual' [ga] and of an 'auditory' [ba] leads to the auditory perception of a [da]</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6: Activations of the emitting patterns in response to an auditory stimulation of type [b], modulated by a visual stimulation of type [g].</figcaption></figure>
according to the principle of combination of activation 3.2.2. If [d] is close enough of the other two, it will exceed with this combination the activation of [b] or [g] alone and be the winner of the competition. That is what happens here.

In this simple experiment, we have shown that this model of a node can discriminate language sounds, and even reproduce an auditory illusion that was impossible to reproduce in a simple manner with previous models based on causal connectivity. However, we are not claiming that this experiment is an accurate description of how the real McGurck effect is produced by the brain. At least, it shows a demonstration of how we can use the model to simulate observable phenomena with a proper choice of parameters.

# 5 CONCLUSION AND FUTURE WORK

Cerebral modelling is particularly important in the medical field, to better understand the possible outcomes of a lesion or a degenerative disease. Causal connectivity [14], although promising, is still lacking a certain biological plausibility. In this work we tried to fill this gap a little, and we managed, doing that, to reproduce a phenomenon that was not modelled before by the other models of this family.

Our approach found its roots in empirical knowledge on neurophysiology and we scaled it on an “intermediary level”: the neuronal populations. This interpretation finds an interesting echo and a beginning of formal justification in the work of Balkenius and Gardenförs [5, 6], who proved that it was possible to produce non-monotonic logical inferences using activation patterns of neuronal networks. The inferences could then be encoded by weighted formulas as in probabilistic logic [7]. This calls for a comparison with the similarity-based inference mechanism as a future line of research.

Regarding the future work, dynamic learning mechanisms are being developed to build and tune the rules using only incoming information. In terms of machine learning, it is an *online* and *unsupervised* learning. We plan to test these learning abilities in two ways. They will be compared to other algorithms on machine-learning benchmarks. We will also design a simple categorisation experimental task and will compare results of the task performance by humans and by the model. Furthermore, using the work of Balkenius and Gardenförs as a starting point, we would like to formally prove that the neuronal subtratum can support high level logic when one reasons in terms of patterns. This would contribute to fill the gap between connectionism and symbolic logic. Lastly, it has been shown in biology that the action of different neurotransmitters modulates the behaviour of neuronal populations (e.g. threshold lowering as in McGurck effect, see 4.2). By modelling specifically this action, we would be able to modulate our system in a systematic manner, while being even closer to biological reality [15].

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