Neural modeling field (NMF) theory mathematically implements the mind mechanisms including concepts, emotions, instincts, imagination, thinking, understanding, language, interaction between language and cognition, the knowledge instinct, conscious, unconscious, aesthetic emotions including beautiful and sublime. NMF provides a foundation for modeling evolution of languages, consciousness, and cultures.
NMF is a multi-level, hetero-hierarchical system [1]. The mind is not a strict hierarchy; there are multiple feedback connections among adjacent levels, hence the term hetero-hierarchy. At each level in NMF there are concept-models encapsulating the mind’s knowledge; they generate so-called top-down signals, interacting with input, bottom-up signals. These interactions are governed by the knowledge instinct, which drives concept-model learning, adaptation, and formation of new concept-models for better correspondence to the input, bottom-up signals.
Here we describe a basic mechanism of interaction between two adjacent hierarchical levels of bottom-up and top-down signals (fields of neural activation; in this aspect NMF follows[2]; sometimes, it will be more convenient to talk about these two signal-levels as an input to and output from a (single) processing-level. At each level, output signals are concepts recognized in (or formed from) input, bottom-up signals. Input signals are associated with (or recognized, or grouped into) concepts according to the models and the knowledge instinct at this level. This general structure of NMF corresponds to our knowledge of neural structures in the brain; still, here we do not map mathematical mechanisms in all their details to specific neurons or synaptic connections. The knowledge instinct is described mathematically as maximization of a similarity measure. In the process of learning and understanding input, bottom-up signals, concept-models are adapted for better representation of the input signals so that similarity between the concept-models and signals increases. This increase in similarity satisfies the knowledge instinct and is felt as aesthetic emotions.
The Knowledge Instinct
At a particular hierarchical level, we enumerate neurons by index n=1,2..N. These neurons receive input, bottom-up signals, X(n), from lower levels in the processing hierarchy. X(n) is a field of bottom-up neuronal synaptic activations, coming from neurons at a lower level. Each neuron has a number of synapses; for generality, we describe each neuron activation as a set of numbers, X(n) = {Xd(n), d = 1,... D}. Top-down, or priming signals to these neurons are sent by concept-models, Mm(Sm,n); we enumerate concept-models by index m=1,2..M. Each model is characterized by its parameters, Sm; in the neuron structure of the brain they are encoded by strength of synaptic connections, mathematically, we describe them as a set of numbers, Sm = {Sma, a = 1,... A}.
Models represent signals in the following way. Say, signal X(n), is coming from sensory neurons activated by object m, characterized by parameters Sm. These parameters may include position, orientation, or lighting of an object m. Model Mm(Sm,n) predicts a value X(n) of a signal at neuron n. For example, during visual perception, a neuron n in the visual cortex receives a signal X(n) from retina and a priming signal Mm(Sm,n) from an object-concept-model m. Neuron n is activated if both the bottom-up signal from lower-level-input and the top-down priming signal are strong. Various models compete for evidence in the bottom-up signals, while adapting their parameters for better match as described below. This is a simplified description of perception. The most benign everyday visual perception uses many levels from retina to object perception. The NMF premise is that the same laws describe the basic interaction dynamics at each level. Perception of minute features, or everyday objects, or cognition of complex abstract concepts is due to the same mechanism described below. Perception and cognition involve concept-models and learning. In perception, concept-models correspond to objects; in cognition models correspond to relationships and situations.
Learning is an essential part of perception and cognition, and it is driven by the knowledge instinct. It increases a similarity measure between the sets of models and signals, L({X},{M}). The similarity measure is a function of model parameters and associations between the input bottom-up signals and top-down, concept-model signals. For concreteness the following text refers to an object perception using simplified terminology, as if perception of objects in retinal signals occurs in a single level.
In constructing a mathematical description of the similarity measure, it is important to acknowledge two principles (which are almost obvious). First, the visual field content is unknown before perception occurred and second, it may contain any of a number of objects. Important information could be contained in any bottom-up signal; therefore, the similarity measure is constructed so that it accounts for all bottom-up signals, X(n),
- L({X},{M}) = ∏n=1..N l(X(n)).
This expression contains a product of partial similarities, l(X(n)), over all bottom-up signals; therefore it forces the mind to account for every signal (even if one term in the product is zero, the product is zero, the similarity is low and the knowledge instinct is not satisfied); this is a reflection of the first principle. Second, before perception occurs, the mind does not know which object gave rise to a signal from a particular retinal neuron. Therefore a partial similarity measure is constructed so that it treats each model as an alternative (a sum over concept-models) for each input neuron signal. Its constituent elements are conditional partial similarities between signal X(n) and model Mm, l(X(n)|m). This measure is “conditional” on object m being present (Perlovsky 2001), therefore, when combining these quantities into the overall similarity measure, L, they are multiplied by r(m), which represent a probabilistic measure of object m actually being present. Combining these elements with the two principles noted above, a similarity measure is constructed as follows:
- L({X},{M}) = ∏n=1..N ∑m=1..M r(m) l(X(n) | m).
The structure of the expression above follows standard principles of the probability theory: a summation is taken over alternatives, m, and various pieces of evidence, n, are multiplied. This expression is not necessarily a probability, but it has a probabilistic structure. If learning is successful, it approximates probabilistic description and leads to near-optimal Bayesian decisions. The name “conditional partial similarity” for l(X(n)|m) (or simply l(n|m)) follows the probabilistic terminology. If learning is successful, l(n|m) becomes a conditional probability density function, a probabilistic measure that signal in neuron n originated from object m. Then L is a total likelihood of observing signals {X(n)} coming from objects described by concept-model {Mm}. Coefficients r(m), called priors in probability theory, contain preliminary biases or expectations, expected objects m have relatively high r(m) values; their true values are usually unknown and should be learned, like other parameters Sm.
We note that in probability theory, a product of probabilities usually assumes that evidence is independent. Expression for L contains a product over n, but it does not assume independence among various signals X(n). There is a dependence among signals due to concept-models:each model Mm(Sm,n) predicts expected signal values in many neurons n.
During the learning process, concept-models are constantly modified. In this review we consider a case when functional forms of models, Mm(Sm,n), are all fixed and learning-adaptation involves only model parameters, Sm. From time to time a system forms a new concept, while retaining an old one as well; alternatively, old concepts are sometimes merged or eliminated. This requires a modification of the similarity measure L;