Contributed by: Paolo Arena and Luca Patanè, DIEES, University of Catania, Italy.

Cellular Nonlinear Networks Basics
Cellular Nonlinear Networks (CNNs) were introduced by Leon Chua in 1988, as large arrays of locally connected simple analog nonlinear cells, with the capability of being digitally programmable through the modulation of the local connections among the cells, under the form of Cloning Templates. A schematic view of CNNs is reported in Figure 1. In particular the state variable of each cell depends on the input, output and state values of the neighboring cells by means of the voltage controlled current sources outlined in Figure 1. Moreover, the output nonlinearity is very simple, a saturation function. In the scheme the relations among the Cloning templates A, B, C, and the circuit parameters are also outlined. Finally, a VLSI implementation of the CNN architecture is presented. Nowadays technology allows to embed inside a single analog, digitally programmable chip, a number of 176 by 144 analog cells, able to realize a lot of functions in real time, assured by the analog implementation.

Figure 1: The Cellular Nonlinear Network scheme and main characteristics

Due to the ever increasing number of different applications, the initial definition of the CNN was subsequently generalized as an array of mainly identical nonlinear circuits called cells, mainly locally connected through static or dynamic (including delay) connections, collected into cloning templates. This phase, being realized in an analog array, takes place in real time, making CNNs a paradigm for the real time study of non linear complex dynamics.

The CNN Universal machine
CNNs can be used as programmable, continuous time, discrete space processing devices where the instructions are represented by templates, while initial conditions, boundary conditions and inputs play the role of operands. Once defined these elements, the results of the operations is obtained after a continuous time evolution, within the array. This is the idea behind the so-called CNN Universal Machine (CNNUM), a further evolution of the CNN architecture, proposed by T. Roska and L.O. Chua, in which both analog and digital circuitry coexist. The adjective “universal” must be intended in the sense of Turing. This statement hides a rather complex definition but the essential meaning is that a Turing (universal) machine is able to realize any conceivable algorithm (recursive function). This is an important result that proves the existence of an algorithm. However, it does not give an answer on how the CNN/CNNUM can perform a desired algorithm. On this basis, the CNNUM is a CNN in which every analog cell is augmented by the introduction of additional local analog and digital blocks. The coexistence of analog and digital blocks, working together, without any data converter placed within them, suggested the introduction of the term “analogic computing”(from the contraction of “analog” and “logic”).

Application of CNNs to model bio-inspired locomotion
CNNs were initially introduced at the main scope to act as real time image processors/computers. Subsequently they acquired the role of paradigms for the study of complex dynamics, including spatial temporal chaos and self-organization. Several experimental results demonstrated that CNNs are able to show the peculiar characteristics of complex systems, like biological neural networks. The emergent dynamical characteristics of lattices of simple non linear units lead to the emergence of new solutions, which are often characterized by order and harmony. Moreover here computation is rather “wave based”, than “bit based”. In this direction, these continuous time spatial temporal dynamical circuits and systems are the paradigmatic mirror of biological neural computation. CNNs, thanks to their programmability, allow to design and build space distributed networks of nonlinear systems (neurons) able to process in real time analog, space-distributed signal flows. Of course, both for the sake of simulation and for the fact that nowadays programmable digital hardware is available at low costs, several model of discrete time CNNs were introduced. As an example, second order, spiking cells were designed and implemented by using continuous time nonlinear circuits, whereas diffusion-like synapses were chosen for the connections among the cells. The designed CNN belongs to the class of Reaction-Diffusion CNNs, able to generate particular traveling waves along the neurons of the CNN lattice: the so-called Autowave fronts. These autonomous fronts were shown to possess the same qualitative characteristics as the signals in neural fibers. The design started from the dynamics of the single cell in the CNN array. This cell was designed so as to show, in the phase plane, a slow-fast, spiking limit cycle, typical of excitable media, like the neural one. The design was further tested against noise tolerance and was found robust enough to allow first a discrete component realization, and finally a VLSI implementation. Autowave fronts travelling through CNN neurons are extremely flexible: in fact their path along the cells can be efficiently controlled by using an input mask, i.e. a sensory input. The final aspect was to design a suitable functional transformation to realize a correspondence among the state variable of the CNN neural cells and the joint variables of the legs in the robot. This task was successfully achieved, realizing multi joint biorobots where locomotion was the visible solution of the neural dynamics designed and realized at the level of the neural circuit of the controller (RD-CNN). An efficient approach consists in pre-designing a series of locomotion patterns by implementing a set of chemical-like synaptic connections among the neurons. In this case, the network topology remains the same, but a different set of template values is uploaded as a function of the particular locomotion pattern to be implemented into the robot. This last approach was called Multi-template approach.

Modelling action-oriented perception through CNNs
Drawing inspiration from perceptual mechanisms of biological systems, and relying once again on the RD-CNN paradigm, a bio-inspired framework for the sensing-perception-action cycle, was recently designed and implemented to be applied, as a first simple example, to the real time control of robot navigation, where the robot task is to move in a cluttered environment, trying to avoid randomly placed obstacles and to reach targets. Such a framework can be divided into functional blocks. The starting point is a sensing block, which receives sensorial stimuli from the environment, dynamically clusters and uses them as initial conditions for a two-layer RD-CNN, which is the core of perception. The CNN parameters are chosen appropriately to generate the so-called Turing patterns, which are here exploited and used to form an internal state representation. The representation of perceptual states under the form of Turing Patterns in CNNs was a working hypothesis, partially inspired by works in Neurophysiology that report on the presence of non-spiking neurons in the cerebral ganglia of some mollusks whose high or low level plateau potentials maintain a specified pattern in front of desired behaviors. Characteristics of the whole perceptual process are: ability to represent different environment situations as internal states; ability to associate a specific action to each internal state; ability to plastically modify these associations thanks to the experience.

Internal states (i.e. Turing Patterns) are the core of the perceptual process since they link sensing to action. They, on the one hand, are the result of the dynamic processing of incoming input stimuli and, on the other hand, represent different ways to interact with the environment. To meet these tasks, we use a CNN as dynamical system and consider Turing patterns as internal states. In particular we use a two-layer four by four Reaction-Diffusion-CNN (with zero-flux boundary conditions) with appropriate parameters to generate Turing patterns. Each pattern is associated with an action by means of a simple reinforcement learning. To perform its task, the robot is provided with no a priori knowledge and learns by means of trial and error, according to the experiments on Classical and Operant Conditioning. The learning is implemented by two mechanisms: an unsupervised learning acts at the sensing block allowing the system to modulate the basins of attraction of the Turing patterns, while a simple reward-based reinforcement learning is devoted to build up the association between Turing patterns and actions. The latter is based on a simplified version of the traditional Motor Map (MM) algorithm. The set of actions to be performed by the robot is not a priori established, instead it is the result of a simple, but effective learning mechanism, which improves the plasticity of the methodology. The sensing-perception-action loop is modeled by using nonlinear dynamical systems like CNNs, exploiting their real-time implementation. The unsupervised learning algorithm has been introduced between the input sensors and the RD-CNN for the dynamical modulation of basins of attraction associated with Turing patterns. Moreover we have designed and used an oversimplified version of the MM, and added a contextual layer to support higher level navigation strategies. In the case of navigation in unstructured environment, the navigation task in a physical space is mirrored into a navigation, in the robot “brain”, through a sequence of basins of attraction, each one corresponding to a particular behavior that has to be performed by the robot, in order to fulfill its mission. An interesting fact is that the cell structure generating wave fronts for locomotion control is structurally equal to that one generating Turing patterns. The two dynamics are obtained simply by a parameter modulation, in strict analogy with biological neurons, which attain different dynamics, although being structurally equivalent. The perceptual architecture was implemented in an FPGA based board architecture for the sake of simplicity and possibility to optimise the structure, but in the near future it is envisaged to have the structure within a whole analog circuit, devoted to generate both the perceptual states and the low level locomotion commands for the robot actuators. Several robotic platforms were also designed and built to act as test beds for the proposed methodology, currently under further assessment and refinement.

References

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P. Arena, A. Basile, L. Fortuna, M. E. Yalcin, and J. Vandewalle, “Watermarking for the Authentication of Video on CNN-UM”, Int Workshop on Cellular Neural Networks and Their Applications, 2002.

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External Links
EU Project SPARK I “Spatio-temporal Patterns for Action-oriented Perception in Roving Robots” http://www.spark.diees.unict.it

EU Project SPARK II “Spatio-temporal Patterns for Action-oriented Perception in Roving Robots: an insect brain computational model” http://www.spark2.diees.unict.it

Cellular Neural Networks on Wikipedia http://en.wikipedia.org/wiki/Cellular_neural_network

Notes
Action outcome of NA_073-1: Summer School On Non-linear Dynamics and Robotics: From Neurons to Cognition