The article, available here, mentions that the authors have "used these chips as a model to study how the connections between neurons strengthen over time, a process thought to be integral to learning and memory, according to a paper published in the Proceedings of the National Academy of Sciences."

Other press coverage is available here. From this article, we learn that "With about 400 transistors, the silicon chip can simulate the activity of a single brain synapse — a connection between two neurons that allows information to flow from one to the other. The researchers anticipate this chip will help neuroscientists learn much more about how the brain works, and could also be used in neural prosthetic devices such as artificial retinas, says Chi-Sang Poon, a principal research scientist in the Harvard-MIT Division of Health Sciences and Technology."

I had two immediate reactions to this work, one good, another not so good.

The first: technically, a very challenging project, and great execution. Much can be learned, in many domains, in trying to bridge the gap between biology and silicon implementations of biological functions. Bravi to the team!

The second reaction: many of the claimed applications of the chip are, to the best of my knowledge, not true. The neuromorphic community has been looking at implementing synapses with a power consumption and dimension orders of magnitude smaller than the ones implemented in this chip. Such an approach (using 400 transistor) for one synapse would not allow any hope for scaling up a system with millions or billions of neurons, and trillions of synapses. Moreover, to be applicable and useful, resulting devices would need to be small. There is no space and power to simulate every aspect of neural computation. The question is: when to stop implementing all details of biological computation. The answer is: When you have the function you need. Do you need to simulate every single aspect of synaptic computation to build a useful application? I do not believe so.

I also believe that the argument that this device would be key in understanding synapses is equally questionable. To approximate a synapse in a digital device, you need to translate analog processes in digital language. This is a (very challenging) exercise in fitting these dynamics in CMOS. What would this tell us on the underlying biology still escapes me, but I leave to the investigators the benefit of the doubt, and I look forward to learn from them what they in turn have learned about biology thanks to this chip. I continue to believe that large-scale software simulations are the primary tools for these sorts of investigations.

Morale: I am sure it has been a challenging task, and much has been learned from the technical standpoint. The research groups should be praised for that, but not for having invented a device that would revolutionize neuromorphic computing, prosthetics, or neuroscience. Of course, I look forward to be contradicted by future publications from this group!

What we were doing was a very early stage in the whole scheme of the research group's project. We were part of this multi-university research group called CELEST (Center of Excellence for Learning in Education, Science, and Technology) that is trying to find out how the human brain works and implement that for real world applications (see BU article Brainy, but so Artificial).

One of the planned future products is a nanoaerial vehicle which is essential a tiny flying vehicle mere centimeters in area. The group foresees this being used in the military where soldiers could throw handfuls of these into unfamiliar territories. The nanoaerial vehicle would be able to fly fully autonomously using optic flow and adapt to the environment without any human intervention. It could map out the entire area and detect where there were people, perhaps even who was a civilian and who was an enemy depending on weapons detection. Its features may not be limited to reconnaissance; depending on how advanced these vehicles become, they could take out targets or act as markers for guided missiles. The possibilities are endless. The group is still far from this goal, but it is a very exciting prospect. What we were doing over the summer was determining whether or not optical flow was a feasible method of navigation for autonomous robots.

Optical flow is the perception of an object's motion due to the object's pixel shifts as the viewer moves relative to the environment. This is how humans and many animals navigate in their environments. Imagine seeing a beach ball while you are at the beach. As you walk closer towards it, it appears to become larger. Also, if the beach ball is not directly in your line of sight, as you approach it, it will appear to move faster on whatever side of the line of sight it originally was from far away. If you are trying to get the ball, as you are approaching it, you will naturally adjust your direction to get the ball directly in your line of sight. For our project, we were trying to apply this navigational technique common in the natural world to autonomous robots.

We used the iRobot Create with a webcam as our platform and ran our optic flow based navigation algorithm in MATLAB. We ran the iRobot Create in a textured environment for optimal optic flow detection.

Our implementation of the optic flow based navigation program successfully traversed through a textured environment (well lit arena with walls covered by randomly generated dots) as displayed in the two videos below. However, as demonstrated by the first video which is running at normal speed, the processing time per frame must be exponentially cut down for optic flow based navigation to become feasible.

Normal Speed

8X Speed

The other main problem we encountered were poor obstacle, wall, and corner avoidance. This was due to the webcam's narrow field of view (55 deg). ( see Obstacle Avoidance Dilemna Diagram)

FUTURE DIRECTIONS:

There are many possible future directions for this project. First and foremost, the biggest issue is the slow image processing time. This is due in part to running the programs through MA TLAB. Implementing the algorithms on a FPGA (Field- Programmable Gate Array) or even ASIC (Application-Specific Integrated Circuit) chip would significantly speed up the computations. Another possible solution would be to write code that would take advantage of GPUs with their parallel processing to do the optic flow computations.

To solve the issues with the narrow field of view, a wide angle webcam could be used. Another possible solution would be to use an array of webcams and then stitch the frames together to form one high quality panoramic image. This would allow the robot to keep obstacles in the field of view and allow the robot to avoid them more proficiently. The same would apply for wall navigation.

Perhaps the most intriguing future direction for this project is to develop a robot q learning system. It would be a points reward system to 'teach' the robot to navigate using optic flow. The robot should be able to eventually avoid obstacles after many trial runs. For example, if the robot is navigating under certain conditions and it runs into an obstacle, it would get a negative point. Whenever the robot makes the correct decision and avoids the collision, the robot gets a positive point. Eventually, after repeatedly committing the same error, it would learn to avoid the obstacle. Ultimately, the robot would learn to adapt on its own. This feature, if successfully developed, would definitely get us closer to having robots that could function and adapt to new environments and situations without human operators.

ABSTRACT

As new technologies continue to develop, more and more robots are replacing humans in situations deemed too dangerous. However, current solutions are not fully automated, requiring offsite human operators for executing basic actions. The ideal solution would be a fully autonomous vehicle that could complete its objectives without any human intervention. In this project, the viability of optical flow based navigation was investigated. Optical flow, or optic flow, is the perception of object motion due to the object’s pixel shifts as the viewer moves relative to the environment. First, motion detection filters were developed and applied to image sequences in MATLAB. Then, they were implemented in optic flow based navigation MATLAB programs for an iRobot Create with a camera to provide video input. The robot successfully traversed through a textured environment but encountered difficulties when attempting to avoid textured obstacles and corners. Experiments were developed to compare the effectiveness of the Correlation and Gabor filters and to find the relationship between increased motion detection ability and processing time per frames. Possible future directions for this project include implementing GPU (Graphics Processing Unit), FPGA (Field-Programmable Gate Array), or even ASIC (Application-Specific Integrated Circuit) chips to speed up computation time, utilizing a wide-angle camera or an array of cameras to get a wider field of view, and integrating a q learning system.

Research Posters

These research posters were presented at the Boston University Research Internship in Science and Engineering Poster Session held on August 12th in the Boston University Life Science and Engineering Department.

Attached is our research paper.

Optical Flow Based Navigation

Attached is our presentation to the Integrated Circuits and Systems Research Group in the Electrical and Computer Engineering Department.

Optic Flow Based Navigation

Let’s begin by considering how the mission of CNS was described at its web site, archived here.  “How does the brain control behavior? How can technology emulate biological intelligence?” These are inspiring questions. Their articulation was visionary and distinctly “ahead of the curve” of recent developments in the ongoing convergence of computation and neuroscience so dear to readers of this blog. Most interestingly, while there are several research labs in the world with cognate mission statements, it is not easy to find comparable language from an academic unit at the grain of a university department with a coherent graduate curriculum. The coordination of talent among the faculty, postdoctoral associates, and graduate students aggregated at CNS was evidently a rare event in academia.

Established as a program offering doctoral and masters degrees in the Fall of 1988, CNS became a department in the 1990-1991 academic year. Its founder and author of the questions that defined its mission statement was Stephen Grossberg, and the only other tenured faculty member at the department’s founding was Gail Carpenter. CNS blossomed by attracting both additional faculty and pioneering graduate students seeking to train in the uncharted interdisciplinary frontiers of neural network modeling. CNS has to date awarded almost 150 PhD degrees, with a few dozen more students in its pipeline. (Although the CNS Department has been decommissioned, its degree-granting program will continue as long as students in good standing progress toward their degrees.) CNS is no longer admitting new students, and prospective applicants are directed to Boston University’s vibrant new  Computational Neuroscience PhD specialization of Boston University’s Graduate Program for Neuroscience.

A lynchpin of the CNS Department’s impact was its unique curriculum of graduate courses, which attracted students from diverse backgrounds, some identifying themselves as neuroscientists and others as modelers or computational scientists. Rather than combining generic neuroscience courses with “straight” mathematics or computing courses (with perhaps examples drawn from neuroscience), the CNS curriculum featured courses that week after week offered tailored expositions of how computational modeling could bridge behavioral and biological data to yield insights on the brain’s ability to control behavior and to point the way to technological emulation of, or prostheses for, the brain’s functions.  These courses covered the domains of vision, learning and memory, audition, reinforcement learning, decision-making, and control of skilled action. The CNS curriculum conferred two important “fringe benefits”: (1) Students learned to construct models across a range of modalities, establishing strong foundations for future research outside their initial areas of specialization. (2) Students helped to cross-train each other and bonded into strong cohorts, because they took not one or two but five, six, or more courses together. Researchers who hired CNS PhDs as postdocs have marveled at how a department whose reputation was built on neural modeling could produce graduates steeped in data, but this was a core outcome of a training program where computational modeling was viewed as a pursuit in service of demanding data, rather than as an activity to be conducted in isolation.

CNS’s approach to research and training was visually summarized by the Outstein logo posted in this article, which is featured on the archival CNS web site and explained in accompanying text, quoted here:

The Outstein was developed by Chris Pribe, CNS ’93, and depicts a combination of the Ehrenstein illusory circle and the outstar neural network design for presynaptic learning developed by Stephen Grossberg. The thickened outer ends of each line represent strength of synaptic connections to neighboring neurons, represented by dots, from a central neuron, represented by the illusory inner disk. The Outstein thus connotes emergent global effects from local processing and the embedding visual illusion is illustrative of the kinds of manifestations of brain processing studied at CNS.

The grand challenge that CNS undertook endures.  Given the level of coordination among researchers that will be needed to unravel the brain’s mysteries, progress requires functional research units of the size of university departments or institutes, rather than efforts headed by individual laboratory heads and their research groups. Research and training in emerging interdisciplinary skills involving mathematics, computation, and the collection and analysis of biological and behavioral data, must be tightly coupled, both administratively and intellectually, through practical training and coursework.

Massively multi-core computers will soon be widely dispersed in academia and industry. As these machines enable the implementation of models with comparable numbers of artificial “neurons” and “synapses” as primate brains, the need for research that bridges neuroscience and technology such as was pioneered by CNS grows ever more urgent. Are facsimiles of the Outstein doomed to wander cached archives of the web forever, like the Flying Dutchman? Increasing numbers of academic units will soon devote themselves to the challenge of fusing engineering with experimental neuroscience through computational modeling; these units will develop intelligent technologies that will both support and help us to understand our human intelligence. The Outstein’s appeal, still real, will not soon fade away.

Unsupervised learning of orientation selectivity maps in a realistic virtual environment
A substantial amount of research has been conducted to show unsupervised learning of oriented receptive field maps from exposure to natural images. In typical scenarios, learning occurs over an extended period of presentation of random patches extracted from a natural image. In this work we sought to test whether receptive fields would develop in an animat wandering in a 3D virtual world. Our neural network consists in roughly three stages: retinal cones, LGN cells and V1 cells. The output from the first stage (retinal cones) was produced by filtering of the rgb image received by an animat with filters whose spectral characteristics correspond to L, M, and S cones. This particular arrangement allowed us to look at learning across multiple achromatic (black, white) and chromatic (red, green, blue, yellow) channels. Note that most of the self-organization work to date has been conducted with a grayscale channel only. For LGN cell center-surround filters, we used self-normalizing distance-dependent shunting equations (Grossberg, 1982), rather than the usual difference-of-Gaussians. For chromatic channels we used cells characterized by dual spatial and cone opponency. For learning we relied on the BCM rule – for Bienenstock, Cooper and Munro, the originators of the rule – with center-surround competition within each cortical hypercolumn.

The animat experienced the virtual environment as shown for example in the movies below.

Example receptive fields learned from various hypercolumns are shown below.

Figure 1. Learned receptive fields. For each channel, receptive fields are grouped per spatial location (many receptive fields are developed at each spatial location). a) Black channel. b) White channel. c) Red channel. d) Green channel. e) Blue channel.

Oriented receptive fields develop in all channels within a few hours of simulation (on a workstation equipped with nVidia GTX-295s GPU). Upon visual inspection it would seem that the receptive fields are not as finely tuned for some of the chromatic channels than for the achromatic ones. This said, more extensive measurements of the tuning properties should be conducted to fully substantiate that claim. In any case, this experiment acts as a good stepping stone toward our next modeling experiment in which we are trying to learn to control saccadic eye movements in addition to learning receptive fields.

For more info, please visit this page.

References
Bienenstock, E.L., Cooper, L. and Munro, P. (1982). Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex. The Journal of Neuroscience, 2, 32-48.
Grossberg S. (1982). Why do cells compete? Some examples from visual perception. The UMAP Journal, 3, 103-121.

For the first round, I have built a pretty simple frequency domain classifier for steady-state visually evoked potential (SSVEP) that makes decisions once each four seconds, outputting via UDP to a Python controller script developed by fellow Neurdon contributor and Python guru, Byron Galbraith. SSVEP is a nice choice for this project because the discrete movements are easy to control and the SSVEP signal is robust enough to not worry about false positives on a regular basis. The goal is to get this working in under two second decisions using canonical correlation analysis, however, we are currently using Simulink for the decoder. Anyone wanting to discuss the horridness of Simulink for real-time signal processing will find a friend here. Let's just say the Python IDLE window on my laptop never closes lately.

Here's a short video to give a better idea of what's going on:

The story, which has echoed across media outlets from CNN to EETimes, goes something like this: "IBM on Thursday announced it has created a chip designed to imitate the human brain's ability to understand its surroundings, act on things that happen around it and make sense of complex data. [....] One of the prototype cores contains what amounts to 262,144 programmable synapses, and the other contains 65,536 learning synapses." The chip has 256 neurons.

Let's check with a bit more care what the claim is in the two main domains, the hardware and the task. At the end of the day, what we want to know is what we can do with the chip, so the main dimensions we need to consider are power (both computing power, and power consumption), and what I can achieve with it (namely, do I do better than what has been already done?).

The hardware: Neurdon readers are educated ones, and should not be caught by surprise. You can just read the very instructive post by Fope, published last August, that provides some good background info on the topic. Brain chips of the kind that IBM has built are neither new, not for sale at Best Buy. The reason? Well, the main one is behind your screen right now: digital chips of the kind that equip your laptop or tablet PC are just so efficient and flexible that the "neuromorphic alternative" is not viable yet. As I was saying, this sort of chips are not new. Starting with the pioneering work of Carver Mead, "spiking" chips that mimic some of the functional characteristics of biological neurons (namely, the nonlinear, low-voltage dynamics of their membrane potential) have been explored for decades, and there are numerous groups leading the change on this framework. For instance, the Johns Hopkins IFAT chip is one of such examples: the 4th generation system (IFAT 4G) will consist of over 60,000 neurons (10 chips) with 120 million fully programmable synaptic connections. Another example comes from Stanford, with the Neurogrid featuring sixteen Neurocores, each with 256x256 silicon neurons in 11.9x13.9 sq-mm. Not to forget a major player in Europe, the FACETS chip (now continuing in the BrainScales initiative) consisting of 200,000 neurons and 50 million synapses. So?

Well, all these initiatives are from academic institutions that do not have the PR firing power of IBM. Neurdons are warned though: this has already been done, at larger scales.

But there is more. The chip designed by IBM, and many (but not all) of the chips that depart from a purely digital approach in calculating action potentials and synaptic changes do have an advantage over power and size over purely digital approaches, but are severely lacking in one major feature that is essential in both research and engineering applications: flexibility. The learning rule implemented in the IBM chip, which is the major learning rule employed in the SyNAPSE project (spike-timing dependent plasticity, or STDP), is one of the many synaptic plasticity rule used by neurons to give brains their amazing, most important characteristic: the ability to learn. The debate over whether STDP is even a "real" rule is still open (see for instance here). Bad news: when you implement STDP as the only synaptic rule, you are bound to exist in the subset of models that use it. Which is limited, and surely not enough to implement a human or cat brain... even aplysias are more complex that that. Again, now Neurdons know...

But wait! There is more. Despite this post won't go into details on power-size and other more subtle comparison of the IBM chip and what's out there (and soon to come) in the purely digital domain, we believe that the winning hardware platform that will enable the next, real weave in brain inspired computing is going to leverage the ongoing multicore revolution, coupled with the introduction of a memory technology that will reduce communication overhead/power losses by bringing memory closer to computation (see this article for more info). The beauty of this prediction is that... it's not a prediction! It is a mere observation of a trend which is under our eye. An upcoming post will discuss more about this topic. A preview: check out this chip by Adapteva, and guess why it is so great.

The task: According to the news stories, the chip "has demonstrated the ability to play (and win) against a human in the game "Pong" and can also read a written letter 7, even when written in various ways." Not much to say on Pong: a Google scholar search gives 3,430 hits when searching for "neural network PONG".
How about recognizing the number 7? Since you are on Google scholar, search for "neural networks MNIST" (4,170 hits). The field of character recognition is at a stage in which handwritten characters (all of them, not only 7...) can be recognized with almost 100% accuracy by specialized applications, taking into account the presence of size and translation variations, etc. Not very impressive either.

What is this hype about? Well, apparently IBM wants to badly associate their name with some keywords: brain, cognitive, chips. It surely does look like their PR machine does a good job, but Neurdons' antibodies help to tell fiction apart from reality.

Biologically inspired recurrent neural networks are computationally intensive models that make extensive use of memory and numerical integration methods to calculate neural dynamics and synaptic changes. The recent introduction of architectures integrating nanoscale memristor crossbars with conventional CMOS technology has made possible the design of networks that could leverage the future introduction of massively parallel, dense memristive-based memories to efficiently implement neural computation. Despite the clear advances given by memristors, the implementation of neural dynamics in digital hardware still presents several challenges. In particular, large scale multiplications/additions of neural activations and synaptic weights are largely inefficient in conventional hardware, leading, among other things, to power inefficiencies. In this paper, we describe a methodology based on fuzzy inference to reduce the computational complexity of such networks by replacing multiplication and addition with fuzzy operators. We use fuzzy inference systems (FIS) to evaluate the learning equations of two widely used variants of Hebbian learning laws, pre- and post-synaptic gated decay. We test this approach in a recurrent network that learns a simple dataset, and compare the fuzzy and canonical implementation. We find that the behavior of the network using FIS with min; max is similar to that of networks that employ regular multiplication and addition, while yielding better computational efficiency in terms of number of operations used and compute cycles performed. Using min; max operations we can implement learning more efficiently in memristive hardware, translating into power savings. This work, partially supported by the SyNAPSE program of the Defense Advanced Projects Research Agency and by CELEST, a National Science Foundation Science of Learning Center, was authored by Massimiliano Versace, Anatoli Gorchetchnikov (Neuromorphics Lab) and Robert Kozma.

I. INTRODUCTION
Recurrent neural networks are biologically inspired artificial neural network models generally consisting of two main components, cell bodies and synaptic weights. The ratio of synaptic weights to cells is usually very high, hence the simulation of large networks is computationally intensive since it involves a large number of computing elements and operations on these elements. In current electronics a tradeoff is required between speed, power, area, and accuracy of hardware. The recent introduction of memristive memory makes it possible to densely store synaptic weight values for recurrent nets in a combined CMOS/memristor hardware architecture, resulting in memory load reduction. In this paper, we introduce a novel method, based on fuzzy inference, to reduce the computational burden of a class of recurrent networks named recurrent competitive fields (RCFs). A novel algorithmic scheme is presented to more efficiently perform the highly repetitive synaptic learning component in hardware. Memristive hardware holds promise to greatly reduce power requirements of neuromorphic applications by increasing synaptic memory storage capacity and decreasing wiring length between memory storage and computational modules. However, implementing neural dynamics and learning laws in hardware still presents several challenges. Synaptic weight update rules and network dynamics heavily rely on multiplication and addition, the former operation being expensive in terms of power and area usage in hardware. In this paper, we explore an alternative mathematical representation of these computations aimed at improving power efficiency. We focus on recurrent neural networks and Hebbian learning rule variants as described by Snider [1]. Biologically inspired neural networks generally consist of orders of magnitude more synapses than cells. Synaptic weights are usually accessed to perform scalar multiplication with pre-synaptic cell activation at runtime, along with learning and synaptic weight updates. We explore methods based on fuzzy inference systems (FIS) to increase efficiency of implementing Hebbian learning on hardware with respect to using the conventional algebraic operations (+;-). In this paper, an adaptive recurrent network is described in which synaptic weight updating is performed using Takagi-Sugeno type fuzzy inference systems [2]. In Section II, we provide some background on fuzzy systems and the recurrent network employed in this study. In Section III, we describe a methodology to redefine two learning equations by using FIS. In Section IV, we simulate the fuzzy and conventional algebra networks and compare their respective behavior in terms of accuracy and computational efficiency. In Section V, we discuss the results.

II. BACKGROUND
In this section, we provide the necessary background concerning memristors, computational complexity of recurrent networks used in this study, as well as fuzzy inference systems employed to augment the efficiency of computations. We focus on a class of widely used adaptive recurrent neural networks termed Recurrent Competitive Fields (RCFs) [3]. RCFs are massively parallel biologically inspired plastic networks, a characteristic that makes them both powerful and at the same time memory and computationally intensive, introducing issues in the efficient hardware implementation of this class of models.

A. Memristors and the Brain
The memristor, short for memory-resistor, is the fourth fundamental two-terminal circuit element in addition to the resistor, capacitor, and inductor. Memristors were predicted based on symmetry assumptions by Chua in 1971 [4], and discovered at HP Labs in Palo Alto in 2008 [5], when certain materials yielded non-volatile resistance similar to the one theorized by Chua and Kang [6]. Memristors are characterized by hysterisis loops in their current-voltage behavior, as well as the ability to stably maintain their nonlinear resistance with extremely low decay rates after power is switched off, measurable in hours, days, or even years. This property makes them useful as nonvolatile memories. The memristive property only emerges significantly at the nanoscale, explaining their elusive nature until present days [7]. Based on current paradigms in nanotechnology, memristors are packed into a crossbar architecture [5]. Such memristor crossbars have been successfully integrated with Complementary Metal-Oxide-Semiconductor (CMOS) enabling the close placement of memristive crossbars along CMOS processors [8]. The further miniaturization allowed by memristors bears the promise of contributing to one aspect of the solution to Moore’s law slowdown by allowing close placement of memory and computation, minimizing power dissipation, while at the same time overcoming the von Neumann bottleneck related to the physical separation of the processor and memory. The density of memristors and their compatibility with existing CMOS technology makes them also suited to implement massively parallel neuromorphic architecture [9]. Synapses in the brain and memristive devices have similarities in their behavior, prompting the idea to utilize them in neuromorphic hardware. Certain aspects of brain dynamics can be viewed as massively parallel dynamical systems, where neurons constantly read and modify billions of synapses. Storing and updating synaptic values based on synaptic plasticity rules is one of the most computationally cumbersome operations in biologically inspired neural networks. Memristor crossbars make it possible to efficiently approximate biological synapses by packing memristive based nanoscale crossbar arrays close to a CMOS layer at densities of approximately 1010 memristors per cm2 [10], [11], [12] (see Fig. 1). Such an architecture can be used to efficiently simulate neural networks due to neural networks’ ability to tolerate an underlying ”crummy” hardware, such as memristive based devices characterized by a large number of defective components [13]. A typical implementation of memristive based synapses consists of dynamical nanowire junctions formed by the crossing of two nanowires separated by a metal oxide. Implementing adaptive recurrent neural networks in memristive hardware have been proposed in [13]. Memristor implementations of instar and outstar networks on neuromorphic computing architectures have been recently studied by Snider [1], [13]. A method for stochastically approximating the behavior of instar and outstar learning using memristors

Fig. 1. (i) Memristor/CMOS hybrid chip architecture with details of memristor crossbar implementing memristor synapses [11]; (ii) hysteris loop in the current-voltage behavior of the memristor [12].

and spiking neurons is described, making use of the densely packed memristor crossbars as the memory for synaptic models.

B. Recurrent Competitive Field
Neural networks are built from modules of the form dy=dt = f(y(t)) + h(Wx(t)), where vectors x(t) and y(t) represent the cells of two subsequent layers of the network, W is the synaptic weight matrix of edges connecting the cells, and f is a nonlinear function (e.g. sigmoidal) applied to each element of its vector argument. Using enough such modules and sufficiently large matrices, one can approximate any continuous function arbitrarily well. Computation along synapses can be interpreted as scalar multiplication, where the strength of connection between the cells is reflected in the value of the synaptic weight connecting the cells. Onesuch network is the recurrent competitive field (RCF). RCFs are a class of biologically inspired neural networks introduced by Grossberg that learn input/output representations via a modified Hebbian learning law [3]. Variations in RCFs describe networks that include feedback pathways allowing a cell’s output to project back to its input either directly or indirectly through off-center on-surround projections. Typically, an RCF cell receives, in addition to its bottom-up input, a self-excitatory connection as well as inhibitory connections from neighboring cells in the same layer. RCFs are able to compress and store activity in short term memory (STM), a property that depends on the choice of the feedback input function. Grossberg showed how to construct networks with stable nonlinear network dynamics with respect to external stimuli [14], [15]. RCFs are described by a system of coupled differential equations, and are a computationally intensive algorithm with respect to their hardware implementation due to the complexity of the dynamics at the cells and the high ratio of synapses per cell. In this study, the network consists of a two-layer RCF, an input layer F1 and an output, or coding layer F2. The layers are connected by bottom-up, or feedforward plastic connections modified by the instar (Hebbian post-synaptic gated decay) learning law, and by top-down, feedback projections modified by the outstar (Hebbian presynaptic gated decay) learning law. Cells in F1 are denoted as xi and the cells in F2 as yi. The RCFs used for simulation in this paper are governed by the following set of equations:

Here Ii denotes the bottom-up input to the cell, the constants A;B, and C determine the behavior of the network, and _ is the scaling factor to influence top down feedback is in all simulations set to 0:01. We achieve supervised learning by boosting the activation of the appropriate coding cell yi, with dim(y) = n, by using a supervised learning term

where ki are constants set to 1 for coding cells and 0 for all other cells. The feedback function f(_) may be chosen to be linear, slower than linear, faster than linear, or a sigmoid. The notation [ _ ]+ denotes max(0; _ ). In this paper f(yi) is chosen to be the sigmoid function

Sigmoidal feedback functions combine the functionality of all three cases by contrast enhancing small signals, storing intermediate signals with small distortion, and uniformizing very large signals. In addition they add a new emergent property, the quenching threshold, the minimum size of initial activity required to avoid being suppressed to zero. The Outstar learning law (Hebbian pre-synaptic gated decay) is used to compute the weights wji of the top-down synaptic connection between cells xi and yj :

Fig. 2. Schematic network diagram of a two layer RCF.

The constants D and E represent the learning rate, and determine the stability-plasticity of the system. Similarly, The instar learning law (Hebbian post-synaptic gated decay) is used to compute the bottom-up synaptic weights wij between cells xi and yj :

The components of variables x; y;wij ;wji are constrained into the range [0; 1] by convention. Training the network consists of presenting an input pattern for t seconds and computing x_i, y_j , w_ij , and w_ ji. In general we use Euler’s method to calculate all variables, while in the fuzzy version of the network w_ ij and w_ ji are calculated by a fuzzy inference system.

C. Computational Complexity of Operations
The computation of the synaptic weight matrcies wij and wji at each time step is particularly power-intensive since its size is the product of the number of cells in the two layers dim(x) * dim(y), and as multiplication in digital hardware requires a number of components roughly proportional to the square of the number of bits in the operands. In addition weights can be modified by synaptic plasticity rules governed by differential equations, the solving of which requires iterative numerical methods such as Runge-Kutta or Euler. The operations AND, OR as well as min and max have computational cost proportional to O(n) the number of bits used, while regular multiplication * uses O(n2) of computational resources per bit. Network size is defined to be the energy required to perform a given computation to a given degree of accuracy. For example, given 16 bits of precision the cost E for various operations is:

E(+); E(max); E(min) =~ 16 (7)
E(OR); E(AND) =~16 (8)
E(*) =~ 162 (9)

Basing a network on computationally cheaper operations would provide substantial energy savings, assuming that:
(i) The new methodology produces networks that are not larger (in terms of power consumption); and
(ii) The resulting networks have a similar expressiveness in approximating continuous functions.
A number of candidates to implement such cheaper networks exist, including morphological neural networks [16]. We investigate fuzzy inference systems (FIS) as an alternative approach that provides similar functionality to the regular (+;*) algebra, while potentially being able to save energy by using the less expensive operations +, max, min, together with fuzzy AND and OR.

D. Fuzzy Inference Systems
Fuzzy inference is a method to create a map for an I/O system using fuzzy logic, fuzzy membership functions, fuzzy if-then rules and fuzzy operations. Typical fuzzy inference systems demonstrate advantages compared to classical methods in the fields of pattern classification, expert systems and intelligent control [17], [18], [19]. In this paper, we use Takagi-Sugeno type FIS with min and max operations to reduce computational load of numerically integrating learning equations. Using fuzzy operators offers power advantages over regular (+;*) algebra. Fuzzy sets are universal approximators of continuous function and their derivatives to arbitrary accuracy [20], thus FIS are applicable to solve learning equations governing recurrent neural networks [21]. Advantages of FIS include the following:
(i) computationally cheap fuzzy operators, e.g. fuzzy AND, OR.
(ii) robustness with crummy data, hardware, or even missing data.
(iii) error tolerance; operations min and max don’t amplify errors.

Points (ii)-(iii) are significant, as the memristive hardware is a nanoscale device with high manufacturing defect rates. Fuzzy inference systems can be challenging to design, as they require ad hoc assumptions on membership functions and rules. In addition the defuzzification step can be computationally intensive.

III. FIS METHOD
We propose a design for a fuzzy inference system to compute the pre- and post-synaptic gated decay learning equations resulting in a potentially more efficient hardware implementation. Iterative numerical methods used to evaluate the differential equations governing learning add to the computational complexity. In our approach, we replace regular multiplication and addition operators with fuzzy operators, and numerical integration with fuzzy controllers to minimize the computational costs of learning. We design a FIS to approximate the theoretical behavior of the learning laws to high accuracy. We consider various fuzzy membership functions and fuzzy rules, in addition to different time steps built into the FIS. Instar and outstar learning laws governing RCFs are framed in terms of ODEs (5, 6). Fuzzy inference systems are universal approximators of smooth functions and their derivatives, hence it is possible to design a FIS that given inputs xi, yi and wij yields output which approximates w_ ij arbitrarily well. To fuzzify synaptic learning, the corresponding ODEs are solved symbolically, and a FIS is designed to approximate the solution surface of the learning equation at time dt, which serves as the built time step of the FIS approximator. This process is called fuzzification of a learning equation, and the resulting FIS can replace numerical methods used to solve the learning equation. In this section, we describe a method to fuzzify the instar and outstar equations governing the RCF.

A. Computation at Cells
In typical RCFs there is an order of magnitude of difference in the number of cells and synaptic weights as #Synaptic weights = 2 _ dim(x) _ dim(y). The equations governing the cells in layers F1 and F2 can have complex dynamics and a large number of terms. Nevertheless, due to the high ratio of synapses per cells, the computation of cell dynamics is less critical from a power budget perspective with respect to weight updating. In this paper we therefore focus on the second operation.

B. Outstar Learning
In an outstar network, the weights projecting from the active pre-synaptic cell learn the pattern at the post-synaptic sites:

This differential equation is fuzzified and solved by a three input, one output fuzzy controller. Each equation governing outstar learning is discretizable, meaning that weight w_ ji only depends on components xi and yj of F1 and F2, so the length of the input and output vectors x and y do not increase the internal complexity of the FIS. As a result, the number of fuzzy rules and membership functions required for accurate learning do not proliferate due to an increase in the number of cells in layers F1 and F2. If we set the same time step dt in the FIS to the time step dt = 0:05s used for canonical Euler numerical integration for RCFs, the number of times the fuzzy controller is initiated is the same as the number of iterations needed for solving the learning equation. If it is sufficient to have the synaptic dynamics at larger time steps, it is possible to use a larger time step dt for fuzzy learning to converge to the desired weights in fewer iterations than needed for numerical methods. The three input one output fuzzy inference system w_ ji = fuzzy(xi; yj ;wji) evaluates the outstar learning law solution at time dt. All inputs and outputs of the FIS are normalized into the range [0; 1]. The constants E and D determining the learning rate are hard-coded into the FIS to reduce the internal complexity of the FIS by decreasing the number of fuzzy rules and membership function evaluations required.

C. Instar Learning
The instar learning equation:

is symmetrical the to outstar learning equation with respect to the two variables xi and yi, so modifying fuzzy inference system to handle instar simply requires switching the two inputs. The same three input, one output FIS is used to implement instar and outstar learning. The two learning rate parameters D and E are hard coded. Computation occurs by changing the order of the input vectors x and y vector inputs, in particular w_ ji = fuzzy(yj ; xi;wij) computes the instar learning law.

D. FIS Parameters
The FIS internal parameters are chosen to maximize computational performance. Internal parameters of FIS include the type and number of input fuzzy membership functions for each input variable, the fuzzy AND/OR rules combining the membership functions, and defuzzification scheme. For performance the number of rules and membership functions are minimized. The FIS approximation improves by the inclusion of additional fuzzy membership functions and rules, but this increases the computational cost of the system. Takagi-Sugeno type FIS are used as opposed to Mamdani type for increased computational efficiency in hardware. For Takagi- Sugeno type inference the output membership functions are set to constant, and defuzzification is done through average defuzzification or weighted average defuzzification. The time at which the learning law ODE solution is approximated based on initial conditions is another parameter of the FIS. We denote this parameter dt as is it analogous to the time step in numerical methods, and determines the learning speed of the system by controlling the number of computing cycles (t=dt) required by the fuzzy inference system to converge to the solution at time t. Common numerical methods for solving differential equations approximate the solution linearly in each small time step. The nonlinear FIS can achieve faster convergence by being able to compute with arbitrary dt and fewer compute cycles as compared to numerical methods. To obtain correct network dynamics the time step dt needs to be accounted for when coupled with the unfuzzified equations of cell activity solved by regular numerical methods.

IV. SIMULATIONS
In this section, we simulate and compare two recurrent networks in two tasks. The first network makes use of regular (_; +) algebra and Euler’s method to compute the change of synaptic weights with instar and outstar learning laws. The second network performs supervised learning through a fuzzy inference system. In the first task, the RCF learns a dataset consisting of images of alphabetical characters. In the second task we apply the FIS learning method to a larger network and compare the accuracy and the computational complexity of the fuzzy and canonical RCFs.

A. Summary of FIS Systems
Three input, one output Takagi-Sugeno type fuzzy inference systems are built with learning rates hard coded (TABLE I). The accuracy of the network is assessed in this case by comparing the rate of convergence of weights in the fuzzy and canonical RCFs to theoretical values, in particular by calculating the difference of synaptic weights at the end of the simulation. The average differences in accuracy in one cycle of the simulation are given in the last column of TABLE I. For optimal performance of the FIS the following design choices were made: (1) internal time step dt, input fuzzy membership function types, fuzzy rules, and output membership functions were varied; (2) piecewise linear or constant output membership functions were selected to boost computational efficiency; (3) average defuzzification was chosen over weighted average defuzzification of output membership functions. FIS with two input membership functions for each input, and constant output membership functions for each fuzzy rule are able to give sufficient approximation of synaptic dynamics for correct classification.
The fuzzy systems described in TABLE I use constant output membership functions. The number of rules is equal to the product of the number of inputs and the number of input membership functions. The FIS are three input systems with two membership functions per input.

TABLE I. Parameters of some Takagi-Sugeno FIS for performing instar and outstar learning. Accuracy is defined as mean square error of the FIS compared to the theoretical values under one iteration of the FIS. If the time step dt is small, then piecewise linear tent membership functions do well in approximating the solution. Trapezoidal membership functions captured the dynamics with a lesser degree of accuracy. In contrast, nonlinear Gaussian membership functions have a stable error rate
throughout all time scales.

B. Character Recognition Network
In this section we test the behavior of two networks in a simple supervised learning paradigm. We construct two versions of the RCF, in the first the synaptic weights are calculated by the Euler method, while in the second a FIS described in the previous section is used to perform the synaptic weight updating.
1) Methodology: The network consist of 51 cells and 1820 synaptic weights. The input dataset consists of 26 binary 7-by-5 pixel input images of the alphabet characters. Accordingly, we use 35 cells x in F1 corresponding to the 35 pixels of the input image. The layer F2 consists of 26 cells y corresponding to the 26 letters of the alphabet. Each yi neuron learns to code for one letter of the alphabet. The initial values of instar synaptic weights wij and outstar synaptic weights wji are initialized randomly in the range [0; 0:1].We initialize xi randomly on range [0; 0:02], and yj are set to zeros. In order to achieve supervised learning, the activation of the cells yi is boosted by the supervised learning term, defined as vector Supj = f(a1; a2; .... ; a26)jaj = 1 and ai = 0 if i 6= jg. The bottom-up input to the cell is a binary value 0 or 1 corresponding to a black or white pixel, respectively. In the canonical RCF network, Euler’s method is used to solve the instar and outstar learning equations with time step dt = 0:05s. The RCF network employing FIS has the same network structure, and FIS described in TABLE I are selected to perform synaptic weight updating. Input letters were presented for 200 computing cycles with the supervised input to the coding cells, interleaved by 100 computing cycles where no input was provided, in order to reset the activity of the network in between letter presentation. The following three tests are performed to assess network behavior:
(i) Presenting the i-th letter in the alphabet as input to F1, verify that cell yi of F2 trained to code it becomes active and other cells do not, i.e. activation of coding cells are near 1 while activation of other cells are near 0.
(ii) Activating cell yi of layer F2 and observing the activation pattern of the pixels of the i-th letter in cells xj in layer F1 due to the feedback-mediated activation.
(iii) Compare the convergence of the synaptic weights learned by the FIS and canonical RCFs through calculating the absolute error ABS[Wij (canonical) – Wij (fuzzy)]
2) Results: The results of tests (i)-(iii) are shown in Fig. 3. Fig. 3 for the FIS using tent input membership functions, and dt = 0:05 (i) shows the performance of the fuzzy RCF network in a character recognition problem. The left image contains the input image, the center image shows the activation of the F1 cells after the input was presented for 100 computing cycles, and the right image shows the reconstructed input in F1 after activating F2. Fig. 3 (ii) has the difference of the outstar weights of the FIS and regular RCF networks after training for t = 10; 000 cycles. Fig. 3 (iii) shows the difference of the instar weights.
The difference of weights the networks are calculated by formula ABS[Wij (canonical) – Wij (fuzzy)] in both cases.
Every FIS listed in TABLE I produce networks that classify correctly. Classification accuracy is important as we are less interested in the explicit synaptic weight values and more in the overall dynamical behavior of the networks. In summary, RCF networks using FIS are able to learn synaptic weights with small absolute error rates, and classify characters correctly. The error in synaptic weights is larger for fuzzy and instar than outstar, nevertheless the classification of characters is correct.

Fig. 3. (i) Left: the pixelated input letter; Center: the activation of yi given letter input R after learning; Right: the reconstruction of the letter R after the activation of y18; (ii) The instar weight matrix maximum error for each letter; (iii) The outstar weight matrix maximum error for each letter.

C. Larger Scale Network Simulation
The purpose of this simulation is to test the speed-up of learning achieved by the reduction of compute cycles by using different time steps dt in the FIS, while applying the FIS methodology to a larger network.
1) Methodology: A simplified two layer RCF is used with one cell in F2 and 5122 cells in F1. Synaptic weights are trained to learn a 512 by 512 grayscale photograph, in which 0 and 1 represent a black and white pixels respectively, and intermediate values represent appropriate grayscale values. In this network, there are 219 synaptic weights and 218+1 cells. The Lena image is used for this simulation. The purpose of this second simulation task is to demonstrate applicability of the FIS methodology to larger scale networks requiring significantly more computational resources. The network preserves the RCF dynamics in F1 while setting y1 = 1 in F2 for all time. We show that by using FIS with larger built in dt time steps we are able to decrease the number of compute cycles required for learning. The synaptic weight computations are performed by FIS, while Euler is used to compute cell dynamics. If the built in time step dt at which the FIS operates is set larger than the one used in calculating the cell dynamics, then the two need to be reconciled. A multitimescale integration scheme is introduced to synchronize the learning times at the cells and synapses. Two nested cycles are used, the outer cycle executes the weight learning by

Fig. 4. Simulating the second task using the Lena photograph as input. The weights produced by the FIS and canonical learning at the end of the simulation at t = 2000 cycles are almost identical.

the FIS with greater dt, while the inner cycle performs the activation at the cells with the smaller time step used by the numerical method at a rate to synchronize the time duration of learning at the cells and weights. Similar considerations were made for other memristive architectures, when using spike-based approximations of network dynamics in conjunction with Runge-Kutta [1].
2) Results: Using the FIS method it is possible to reduce the number of compute cycles required to perform learning in RCFs. This is due to the fact that the numerical method implements a linear approximation of the solution at each step, whereas FIS uses a nonlinear (or piecewise linear) global approximation of the solution surface.

D. Results on Computational Complexity
The computational burden of fuzzy inference and regular synaptic weight updating are compared by determining the number and type of operations needed, and the computational complexity required to implement them. All fuzzy inference systems described in TABLE I perform identical number of fuzzy operations. In TABLE II, the number and computational complexity of operators used under one iteration of the canonical and FIS synaptic weight update method are given. The computational complexity is compared for numerical representations with 4, 8, 16 and 32-bit precision. For the comparison the time step dt is assumed to be the same in the FIS and forward Euler integrator (See Fig. 6), and an identical number of computing cycles are used. The numerical representation in hardware is planned at 16-bit

Fig. 5. Comparison of regular learning and fuzzy learning weight convergence. Top: canonical solution of outstar learning with dt = 0:05s. Red line represents the input pattern present at the given cycle. Blue indicates the rate of convergence to the given pattern. Bottom: Fuzzy learning with Blue: dt = 0:05s; Magenta: dt = 0:25s; Purple dt = 1s. All have the same learning rate.

TABLE II: Number and type of operations required for synaptic weight updating.

precision. In the case of 16-bit precision there is approximately 51:22% improvement in efficiency, while with 32-bit precision the same number is close to 65:43% when using the FIS method with the same time step. For 8-bit precision the FIS methodology only improves computational efficiency by 23:81%, while using 4-bit precision the FIS increases computational cost by 27:27%. The number of compute cycles required can be reduced by increasing the time step dt to further reduce computational burden.

Fig. 6. Computational complexity of synaptic learning as a function of number of synaptic weights. Both regular and fuzzy versions use time step dt = 0:05 for integration. Thick graphs represent FIS performance, while thin graphs show forward Euler performance. The dashing corresponds to numerical precision; non-dashed 32-bit; long-short dashed 16-bit; shorter dashes 8-bit; dotted indicates 4 bits of accuracy.

E. Simulation Issues
(i) When simulating fuzzy inference systems on digital hardware, the execution of fuzzy operators is inherently slower compared to the built in multiplication and addition operators, so runtimes of the simulation are not an indicator of performance.
(ii) The learning rates are hard coded into the FIS, so if we would like to change the learning rate we need to update the fuzzy controller.

V. CONCLUSION
In this paper it is shown that fuzzy inference systems can be useful in providing an alternative hardware implementation for widely used biologically inspired, plastic neural networks. These networks can be efficiently implemented in the CMOS/memristor hybrid hardware architecture. Using the FIS methodology it is possible to significantly reduce the computational complexity of the proposed memristive hardware starting at 16 bit numerical precision by replacing the regular addition and multiplication operations by fuzzy operators. Future work may focus on generalizing this method to other learning laws of the Hebbian family, and performing comparison of noise tolerance to simulate
crummy nanoscale hardware for more complex tasks.

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Computational and theoretical neuroscience has emerged as one of the leading methodologies in helping understand the activity in individual neurons and how neurons cooperate in networks. Computational neuroscientists develop mathematical and computer descriptions of neurons and neuronal networks and then let the computer simulate their activity, step by step. This allows measurement of the state of an entire system, control over all of its parameters, and observation across scales that are more easily interpretable.

However, models are simplified versions of reality. When constructing a model, what is essential and what, if anything, can be pared away, while still allowing a better understanding of the function of the system? Occam's razor would maintain that the best model is the one that can replicate activity of the real system using a minimum of assumptions.

To illustrate how a model can be used to shed light on a biological phenomenon, computational neuroscientists often focus on a single phenomenon of interest. One area that has received a lot of attention in this field is that of neuronal oscillations, or periods of synchronous neuronal firing at varying frequencies. Functionally, oscillations have been proposed as a solution to how neuronal networks combine the different pieces of information of a single object. Oscillations are indeed readily observed in the brain, with the total power of these oscillations distributed in a characteristic way over its range of frequencies.  Using a simplified but biologically realistic neuronal network, Neymotin et. al. was able to demonstrate that in the piece of neocortex simulated, the oscillations emerged largely from a specific layer. Further manipulations revealed a homeostatic mechanism in the model.

Graphing intracolumnar wiring, where circle size represents the number of cells in the population, and line thickness represents connection strength. This approached revealed a central role for excitatory cells
in layer 2.

The model was simplified, and was tuned only to produce realistic firing rates and to avoid pathological spiking activity. Remarkably, from the model's simplified structure emerged a biologically realistic spectrum of rhythms. Cells fired in equal measure across the frequency spectrum when cells were not connected. When they were connected, frequency peaks emerged in the theta/alpha spectrum in the excitatory cells, and in the gamma spectrum among inhibitory cells. Previous researchers have suggested that gamma oscillation work with slower theta oscillations in a multiplexing mechanism that allows information to be shared between different modalities and internal sources in order to integrate them into coherent representations. In telecommunications, multiplexing is the mechanism by which many telephone conversations can be carried over the same telephone wire, for example. In this model, subsets of cells are fired on a particular gamma cycle superimposed on a theta/alpha cycle, the theta/alpha cycle being a physical substrate for a representation composed of information from many different subsets of cells. This multiplexing model is consistent with the physical processes that came about in simulation.

Local Field Potential recordings from left medial prefrontal cortex of an awake rat, compared to simulation. Addition of hubs did not change the general shape of the frequency spectra -- evidence of homeostatic
mechanisms in cortex.

The group had hypothesized that areas of high neuronal density and connectivity might serve a control function for areas around it. In order to better visualize cortical structure, they graphed the structure of a section of the neocortex, taking note of neuron density of excitatory and inhibitory cells separately, as well as connection strength between layers and between columns. Using this graph-theoretic approach, cortical layer 2/3 (the neocortex has six layers, layer one being the one closest to the skull) was revealed to have the greatest cell populations and strongest connections. But determining whether this area played a special role in determining neocortical dynamics was not straightforward. The paper is critical of the ablation experiments often used to determine causal relations in the brain, as removing large portions of cortex as is typical in ablation experiments is likely to bring about a new dynamical regime. The addition of “hubs” to the network offered a gentler way to perturb the network. Hubs were cells that had the three times the number of inputs and outputs as regular cells. Adding hubs to layer 2/3 greatly increased the power in the network, but adding hubs to other places did not do the same, supporting the hypothesis that the layer might serve a control function.

Besides a system to coordinate oscillations, the brain needs a way to return to a dynamic balance after a disruption. Adding drive to the model – which would mimic the deployment of attention, for example – increased total power, but did not change its spectral profile, which may demonstrate a homeostatic mechanism in the neocortex that is intrinsic to its structure. In another manipulation, synaptic delays were increased and the power spectrum still remained the same. The paper suggested that these results would be testable in-vivo, through behavioral attention studies, and in-vitro, by cooling brain slices to induce increased synaptic delay.

The paper highlights the importance of visualization methods to parse out important correlations in what might seem like a meaningless jumble of activity. Neuronal density and the connections between neurons were visualized using graph theory. Graphing of correlations of frequency fluctuations through time revealed temporal coupling of gamma and theta rhythms. The paper suggested visualization of multiple aspects of cell typology, wiring, and dynamics at different scales. When detailed simulations are not feasible, creative modeling can help link different levels of detail (e.g., between molecular, neuronal, and network levels). Creative visualization of various types of activity can lead to better intuition about brain processes, which can lead to novel hypotheses for further experimentation and simulation.

References
Neymotin SA, Lee H, Park E, Fenton AA and Lytton WW (2011) Emergence of physiological oscillation frequencies in a computer model of neocortex. Front. Comput. Neurosci. 5:19. doi: 10.3389/fncom.2011.00019

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