First of all, for those not familiar with SPICE, I recommend downloading the free LTSPICE IV by Linear Technology. It is a light weight analog electronic circuit simulator, and the code written here will be compatible with it. The memristor model  is a subcircuit  comprising of several ideal components and a couple of math formulas.

Being an element with memory, a memristor requires a state variable w, which by normalization can be assumed to lie in the interval [0,1]. Let’s assume that the time derivative of w can be written in the separable form

dw/dt = a*f(w)*g(V),

where a is a constant, f and g are some functions and V is the voltage across the memristor. The window function f controls the way w changes in different parts of [0,1]. In our model f is hard-limited to the interval [0.05,0.95]  and satisfies

f(w) = 1-(2w-1)^2.

In other words the rate of change of w decreases as w tends to its minimum or maximum value. Note that without the hard-limiting the rate of change would go to zero, which would effectively make the memristor’s memory infinite. Let us now concentrate on the other function in the differential equation, namely g(V). The memristor investigated in [Yang et al.] programs very nonlinearly with respect to the voltage V, so  let us model it by a sinh-function as

g(V) = sinh(b*V).

Now, the trick to implement the dw/dt -law in SPICE is to model w as the voltage over a capacitor with capacitance, say, 1F. The change in w is then yielded by a voltage dependent current source (symbol G in SPICE) which adds or removes a certain amount of charge to w. This certain amount is, in turn, determined by computing the functions f and g.

In [Yang et al.] the I-V characteristics of the device were approximated by

I = w^n*c1*sinh(d1*V)+c2*(exp(d2*V)-1),

so let us use that expression as well. The implementation of the I-V characteristics in SPICE is straightforward: just add a voltage dependent current source with this value between the terminals of the device. The real problem is now to find appropriate values for the constants a,b,c1,c2,d1,d2 and n, and here we would need a lot of data to work with. In the following I use values that give a similar I-V curve than the one obtained from experiments. Altogether, the SPICE model hpmemristor.lib looks like

.SUBCKT hpmemristor P M PARAMS:
+ a=25 b=8 c1=9 c2=0.01 d1=2 d2=4 s=1u n=4 p=1 lmin=0.05 lmax=0.95
*State variable:
Gsv 0 w value={f(V(w),V(P,M))*g(V(P,M))}
Csv w 0 1
.IC V(w) 0.05
*Output:
Gmem P M value = {s*((V(w)**n)*c1*sinh(d1*V(P,M))+c2*(exp(d2*V(P,M))-1))}

*Auxiliary functions:
.func sign2(var) = {(sgn(var)+1)/2}
.func trunc(var1,var2) = {(sign2(var1-lmin)+sign2(var2))*(sign2(lmax-var1)+sign2(-var2))/2}
.func f(var1,var2) = {trunc(var1,var2)*(1-(2*var1-1)**(2*p))}
.func g(var) = {a*sinh(b*var)}

.ENDS hpmemristor

Pretty self-explanatory, right? Note that Gsv and Csv are the current source and capacitor needed in the state variable, respectively, while Gmem is the actual “output” current source of the memristor. More specifically,

Gsv 0 w value={f(V(w),V(P,M))*g(V(P,M))}

means that there is a current source between 0 (ground node) and some node called w, and its value is f(V(w),V(P,M))*g(V(P,M)) amps. The value  f(V(w),V(P,M))*g(V(P,M)) denotes the RHS of the differential equation, the constant a has been embedded in the function g, and f needs for a technical reason two variables (this has to do with limiting ourselves to [0.05,0.95]). V(x) gives the voltage of the node x respect to to ground and V(P,M) gives the voltage between the terminals of the memristor. Oh yes, one more thing: .IC V(x) 0.05 means that the initial value of w is assumed to be 0.05. Once you get the model running in LTSPICE, you should get an I-V curve looking something like

That’s it. With different choices for functions f and g (and different parameters) you can model all sorts of memristive devices.

But if the cell acts, after this fashion, as a whole, each part interacting of necessity with the rest, the same is certainly true of the entire multicellular organism: as Schwann said of old, in very precise and adequate words, “the whole organism subsists only by means of the reciprocal action of the single elementary parts.”  As Wilson says again, “the physiological autonomy of the individual cell falls into the background… and the apparently composite character which the multicellular organism may exhibit is owing to a secondary distribution of its energies among local centres of action.”  It is here that the homology breaks down which is so often drawn, and overdrawn, between the unicellular organism and the individual cell of the metazoon.

Whitman, Adam Sedgwick, and others have lost no opportunity of warning us against a too literal acceptation of the cell-theory, against the view that the multicellular organism is a colony (or as Haeckel called it, in the case of the plant, a “republic”) of independent units of life… Hofmeister and Sachs have taught us that in the plant the growth of the mass, the growth of the organ, is the primary fact, that “cell formation is a phenomenon very general in organic life, but still only of secondary significance.”  “Comparative embryology,” says Whitman, “reminds us at every turn that the organism dominates cell-formation, using for the same purpose one, several, or many cells, massing its material and directing its movements and shaping its organs, as if cells did not exist.”…

…Discussed almost wholly from the concrete, or morphological point of view, the question has for the most part been made to turn on whether actual protoplasmic continuity can be demonstrated between one cell and another, whether the organism be an actual reticulum, or syncytium.  But from the dynamical point of view the question is much simpler.  We then deal not with material continuity, not with little bridges of connecting protoplasm, but with a continuity of forces, a comprehensive field of force, which runs through and through the entire organism and is by no means restricted in its passage to a protoplasmic continuum… As Whitman says, “the fact that physiological unity is not broken by cell-boundaries is confirmed in so many ways that it must be accepted as one of the fundamental truths of biology.”

—D’Arcy Thompson, On Growth and Form: The Complete Revised Edition, pp. 343–345

The last paragraph is especially appropriate in light of a recent publication that gives evidence for a possible interaction between cortical network activity and its global, “endogenous electric field.”

The ARM processor is the most widely used 32-bit architecture in terms of units produced. Originally conceived as a processor for desktop personal computers, the relatively simple, low power ARM processors have become dominant in the mobile and embedded electronics market as relatively low cost and small microprocessors and microcontrollers. As of 2007, about 98 percent of the more than one billion mobile phones sold each year use at least one ARM processor.

As Furber was trying to implement processes such as associative memory, he found himself reinventing neural networks. “In the end I threw in the towel: ‘Blow this, what I’m interested in is neural networks”. The wise decision was then to read about it, and build a simulator for them.

It turns out that the path that Farber is taken in the design of the so-called SpiNNaker architecture is a wise one: as opposed to design “rigid” neural architecture in hardware, the choice is to use software to be as flexible as possible in terms of connectivity and neural dynamics. This comes at a cost of size a power, however: the post does not give much details, but chances are that such a system, based on clever, but still traditional, microprocessors, will be relatively heavy and power hungry.

In any case, this is an interesting post and surely worth a read.

Much of the recent interest in HP’s finding the “missing” memristor has been based on the presumption that it is correctly interpreted as the “4^th fundamental circuit element” after the resistor, capacitor, and inductor. The original argument from Leon Chua was that there were four ways to link the variables of charge and current to the variables of flux-linkage and voltage. The resistor, capacitor, and inductor represent three ways to link these variables while the memristor was postulated as representing the “missing” link connecting charge and flux-linkage. This may seem like a reasonable argument at first glance but it is not without flaws.

For example, the argument assumes that charge, current, voltage, and flux-linkage are the only electrical circuit variables. This is actually not the case. Just as current is defined as the time derivative of charge and flux-linkage is defined as the time integral of voltage, higher order electrical circuit variables may be defined in terms of higher order time derivatives or integrals of current and voltage. For, example the switching speed of an electrical signal may be represented by the time derivative of voltage (dv/dt) and a new (fifth?) circuit variable may be defined in terms of charge vs. dv/dt. Thus a 4-element circuit theory may still be incomplete.

A better framework may be formed from a paper co-authored by Leon Chua in 2009 (link) presenting an expanded  view of circuit theory based on dynamic systems analysis. included in this framework are memristive systems, memcapacitive systems, and meminductive systems in which resistors and memristors both belong to the memristive system type (the resistor is the special case where the time derivative of the system state is zero and the memristor is the special case when the time derivative of the system state is equal to the current).  Thus, in a similar manner in which ice and water each represent different forms of H₂O, resistors and memristors each represent different forms of memristive systems.  But water and ice are usually not distinguished as different “fundamental” substances but rather different forms of the same substance. It is in this sense that the idea of the memristor as a new fundamental circuit element is likely to be wrong.

Ralph Etienne-Cummings, from Johns Hopkins University, was recently a guest at ICCNS 2010 in Boston, where he gave a very interesting overview of the state of the art in neuromorphic computing. In their lab, they design biologically Very Large Scale Integrated (VLSI) technology used to extract information about the environment, study how these systems can be hosted onto robots, and develop brain-machine interfaces and neural prosthesis devices, specifically spinal and cortical prosthesis devices and robotic systems to restore function after injury and for human augmentation.

Ralph is on CNN, explaining how neuromorphic chips mimicking locomotion can one day help patients regain their ability to walk. The video is available here.

Some highlights that I found intriguing include:

• Human players have an advantage because they will buzz in immediately after a question is read, even before they’re confident they have an answer. Watson, on the other hand, will always wait until its confidence has reached a certain threshold. In other words, people may have some meta-algorithm to gauge how likely they are to find an answer at they same time that they search for it.
• Watson’s main advantage is its lack of emotional involvement in the game. I’ve also heard somewhere that some of the best baseball players are those who think so highly of themselves that a bad play doesn’t distract them and thus doesn’t start a string of mistakes.
• Watson overcomes a number of implied meanings in the clues by substituting candidate subjects into the phrase and re-running that search against its database. While this method is statistical rather than grammatical, it may capture something more fundamental how we resolve lexical ambiguities as well.

In the end, however, “[research lead] Ferrucci’s main breakthrough was not the design of any single, brilliant new technique for analyzing language. Indeed, many of the statistical techniques Watson employs were already well known by computer scientists. One important thing that makes Watson so different is its enormous speed and memory.” If there is a missing key principle in our understanding of intelligence, then it seems unlikely to expect Watson to uncover such an idea. On the other hand, a question-answering computer expert is a much more general tool than one that plays chess, and I hope to hear results regarding their stated goal of a medical advisory system soon.

The New York Times – What Is I.B.M.’s Watson? – June 14, 2010

Thus, if memristance is actually a required property to simulate neurons, it may be useful to attempt to find memristive systems models for floating gate memory cells. In fact it is not too difficult to show that floating gate memory cells may indeed be modeled using an extended memristive system approach (see link).  In this case the memristive systems approach of Chua may not have to wait for the development of RRAM or future neuromorphic systems to be developed but may be applied to current neuromorphic systems designs based on floating gate cells. In addition, it may not be unreasonable to state that, due to its connection to the floating gate cell, the memristive systems market is currently a multi-billion dollar industry which will only be extended as 2-terminal RRAM gains traction.

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