The Math Underlying Natural Dog Training

(This is a long article to demonstrate how recent science validates the theory of Natural Dog Training as first articulated in the 1980’s.)


Sometimes critics ask for the math that substantiates the theory that underlies Natural Dog Training (emotion=attraction—-feelings=resistance) which is a little tough given that I’m not a mathematician. And the request is also unreasonable because if the logic is there then the math will follow. Also, mathematics is a description, unless that is someone is arguing that the dog is doing the math mentally or there’s an actual computational module running an algorithm somewhere within the dog’s brain. Whereas unlike math, the theory of emotion as an immediate-moment “force” of attraction—with feelings being the processing of resistance-to-attraction—is an explanation and it’s testable in terms of being able to build a model that identifies what is going on within the dog’s mind that can then be correlated, or not, with actual behavior. (some of these correlates I have woven into this article.) And if the immediate-moment manner of analysis can logically encompass the evidence more comprehensively than other systems, especially without having to use Abracadabra magical leaps of logic such as: dogs-can-pick-up-human-cues-from-having-lived-with-humans-for-thousands-of-years—-then it deserves consideration. My hope is that the logical sense of NDT will eventually inspire a mathematician to provide a mathematical description for it and therefore I was quite delighted to hear of a mathematical treatment of the hunting behavior of wolves. It was described to me by Lee Kelley this way: “A dog trainer in Spain recently authored a paper (with two computer scientists, and the blessing of Ray Coppinger) showing that pack hunting behavior can be successfully described using a computer model involving just two simple rules: proximity to prey and proximity to pack members.”


The conclusion of the simulation:

  1. “No Centralized agent is needed”
  2. “No Communication is needed for a successful hunt.”

“Wolf-pack hunting behavior are emergent patterns resulting from the combination of two simple rules applied by each individual. Move towards the prey until a critical distance is reached. When close enough to the prey move away from each other.”


“Successive positions (of pack members relative to the prey) describe a circle.”

“Agents do not use one-to-one communication. All communication is based on sensing the external state of nearby agents.”

In my view, this mathematically validates what I wrote in “Natural Dog Training” to explain the phenomenon of collectivized action in both wolves and dogs. There is no other system of behavior consistent with this mathematical description of hunting. I articulated this with the term “harmonic pathways of learning.”

“Natural Dog Training” First Edition 1992

p. 26: “Circumstances in the hunt are going to be changing constantly. And when I talk about flexibility, I don’t just mean the individuals ability to react to change; I mean that all the members adjust to change as a group. This kind of collective coordination is the  bedrock of sociability. Normally, this kind of coordination might be thought to fall more under the realm of communication, learning, and intelligence rather than of instinct.”

  1. 65: “Wolves don’t learn directly how to cooperate with one another or to target vulnerable prey animals through random trial and error, not even through the consequences of their efforts. In reality, learning has been carefully pre-scripted along the harmonic pathways.”

In other words, no matter how the hunt may unfold, or what changes of circumstances might occur, everything the pack does must fit a script, it has to conform to certain rules; i.e. be in harmony with each other, which translates ultimately into forming a circle.

NDT p.26: “The group has polarities….one for each member and you can visualize its structure as a wheel with spokes radiating outward from a hub. For each individual to be balanced, it needs to be in position along a spoke. Once aligned, the group can come into sync and the wheel is ready to roll.”

Now the main argument I was constructing in NDT was not the specifics of how wolves hunt large dangerous prey, which by the way is a subject in which I am not an expert. I am however an expert on police dog training and I believe that training a dog to search for criminals, then hold a criminal at bay who surrenders, and then apprehend a criminal by biting if he resists or tries to escape, is exactly the same phenomenon as wolves searching, chasing, cornering and if possible bringing to ground a large, dangerous prey animal. The main argument of NDT is not about how wolves hunt, but rather that the fact that wolves evolved so that they hunt as a coordinated group, is also how they configure themselves in the pack, and this social flexibility accords a behavioral plasticity which ultimately granted human beings access to the canine mind so that domestication was possible. Hunting comes first, social follows from that, and all learning fits into the template which governs both. In other words, how wolves evolved to deal with resistance in the hunt, is how they process resistance in the pack. Being social is based on the hunt, rather than the other way around, this is the key to understanding the canine mind.

NDT p. 26: “The premise that the dog is linked to the wolf is nothing new. What is newly being presented here is the idea that sociability is a by-product of, and is dependent on, the prey instinct. Since killing large animals makes advanced social living possible, it makes sense to look for a connection between the higher levels of social development and the prey instinct. Foxes and coyotes, for example, which almost exclusively hunt prey smaller than themselves, do not exhibit this kind of elaborate social structure.”

The consensus view in 1992 remains today what it was then, that the social dynamic is separate from and antecedent to the hunting dynamic. This kind of thinking is why mainstream behaviorists and biologists believe that domestication came first, and then the dog’s capacity to work for man was selectively bred in thereafter. (See Mark Derr’s book: “How the Dog Became the Dog” for an opposite interpretation.) The science on cognition still doesn’t understand that dogs are attuned to where humans point because this is akin to selecting the vulnerable prey animal from a herd, around which the group then focuses their collective emotional charge. Science in its search for higher cognitive processes to explain complex behavior thereby misses the primal emotional dynamic which is the true organizing principle. (The one manifesting the strongest feeling for the prey becomes the compass needle pointing “north.”) And the so-called shift in thinking from the old to the new definition of dominance has only been to recast the mechanism from one of social status, to one of control over various resources. This is more fluid perhaps, but it still is an Abracadabra insertion of an understanding of causes and effects and it’s still based on a control model, i.e. one individual trying to control something about another, just as the old view of hunting was misconstrued as a leader controlling followers. What modern behaviorism doesn’t yet grasp is the nature of the dynamic that this mathematical simulation describes; i.e. Temperament Is a Circle.

When I conduct a seminar, at the chalkboard (and with all due apologies to moose and wolves) I draw the moose at the center of a circle with the wolves configured equidistantly around it. I will next go on to say that the mind of the dog evolved so that this circle or group dynamic is the basis of their personality traits just as much as it is their manner of learning.

As we can deduce from this mathematical simulation, in the hunt results don’t matter, only acting according to two underlying “rules.” This means that tangible reinforcements don’t shape how wolves learn (and which means that tangible reinforcements aren’t the fundamental by which dogs learn.). Achieving a circle is what counts. Even if every hunt fails, they will still learn to act via this predetermined template. And it’s also not logical to believe that this circle-as-template only applies to hunting. Since hunting is what makes the unique and complex social life of wolves possible, we should expect that the circle-as-template is the organizing principle to all phases of their life. It can’t just be that the most important thing wolves have to learn about, bringing a large, dangerous prey animal to ground as a coordinated group, without which they can’t sustain their complex style of living and the communal style of raising the young, is the only thing that they have to learn that conforms to an underlying pattern. Rather, a set of rules, or template as I prefer, would be more fundamental to learning than actual reinforcements. We should expect to find that all aspects of the canine makeup, from how they learn as well as the phenomena of personality and sexuality, and the so-called access-to-resource phenomenon, correspond to a circle as well rather than to intellectual precepts such as an understanding of cause and effect, and access to some “resource” relative to a competitor’s point of view.

At my chalkboard I draw four wolves confronting a moose, with three of them next deflected from a straightforward approach to the flanks and the hind end of the prey. If they can get it circling, whirling around in place, it will tire and may very well grow confused from being constantly attacked from its blind side. It will become susceptible to being brought down.

Because math is but a description we still require a functional mechanics, the actual implementation, unless that is, as I said before one believes there is a neurological module computing the math in the mind. But this isn’t likely because getting close and encircling the prey isn’t the whole story. That doesn’t complete the task. The wolves still have to close the gap, make contact and bring it to ground. Searching, chasing, encircling and bringing prey to ground are all components of the hunting dynamic and we can immediately see that these two rules are not enough to encompass its full scope. All aspects of the hunt, and even social behavior and structure should derive from this same dynamic as well. I also want to reemphasize that it’s only in the human intellectual mind that there is a separation between hunting and social, that the emotion and stress experienced in dealings with their prey are different from dealings with their peers. That is an arbitrary line of demarcation.

Now is there a term that allows us to go from describing to explaining so that we can exit the digital realm of the mathematical zeroes-and-ones that empower this computer simulation and enter the realm of flesh and blood? Yes, there is a term that takes us from behavior in the abstract to a concrete dynamic that is actually transpiring within the animal’s body and mind and from which a model will follow logically. And we can do so without invoking any human thoughts and projecting these onto what an animal is doing. ATTRACTION. When one watches the simulation of small dots chasing a large dot, the one and only thing that can be said with accuracy and without putting any intellectual, abracadabra spin on the ball is that the wolves are attracted to the prey. And immediately this term of flow begets its correlate, RESISTANCE to attraction.

Now let’s review the hunting simulation in terms of attraction and resistance.




So wolves are attracted to the prey, and when the prey runs it conducts the emotion of attraction and the wolves can not only move at full speed but remain next to each other as well. This brings them in close to the prey, but not too close. Then, when configured as a circle about the prey, the prey has nowhere to run and becomes static. The static like pressure induced by the prey’s resistance, polarizes them relative to each other and they spread out because the prey is no longer conducting their state of attraction. They don’t know they are forming a circle, they are feeling where the experience of pressure is at its lowest relative to the prey and to their peers. Their bodies and minds are organized to execute a circle and the same attraction and resistance dynamic which guides their interaction with their prey, is also guiding their interactions with each other. They feel the same pressure towards each other that they feel toward their prey. These are not two different kinds of pressure generated by two different behavioral processes, they are the same dynamic. Another way of saying all of the above is that a feeling of resistance computes an angle of deflection.

“Successive positions (of pack members relative to the prey) describe a circle.”

Temperament as a circle is divided by two axes, the Direct/Indirect “equator,” and then this is transected by the Active/Reactive “longitudinal” axis so that we end up with four quadrants, each populated with two specific traits:


These traits compute for specific roles in the hunt, as well as specific personality traits in the pack. The wolf dealing with the head and horns of the prey is at the Direct/Active pole, the one typically defined as Alpha. The one at the hind end is at the Reactive/Indirect pole, the quintessential Omega. However, when the moose spins, each individual flips poles depending on which end of the moose it is now dealing with. Some can take it some can’t and get out of its way and then try to recover as quickly as possible. These roles are not fixed and this emotional motility means that as a group they can sustain contact. In the hunt this flexibility with an individual flipping from one pole to another happens rapid fire in mere seconds. But the same fluidity is organizing the social structure, it’s just that the migration through the social circle takes much longer, sometimes over the course of a lifetime, although sometimes quicker over a so-called resource, and this has therefore been misidentified as a dominance hierarchy of social rank (the old view), and recently control over access to resources (the new view). But both of these views miss that Temperament-as-a-Circle is polarizing each individual to a specific blend of the four aspects of personality: (D/A)–(I/A)–(R/A)–(R/I); temperament with a lowercase “t” is an individual occupying one particular quadrant of the circle. This is not learning in the sense that is commonly understood, and it’s not working out who controls who or who controls what, rather it’s a process of emotional formatting. Each individual’s deepest level of stress, acquired from the experience of intense resistance to a state of emotional attraction, tunes each individual to an available slot relative to its peers. The emotional states of being variably active or reactive, direct or indirect, translate into angles of deflection. From the vantage point of any given polarity the world looks slightly different, in such a state they each perceive and interpret their environment slightly different from their peers so that as an ensemble they will act in a collectivized and syncopated manner. They all want the one big thing, bringing the prey to ground, but they each will come to have an individualized focus on a different part of its body, a loci, as the focal point of their individual efforts. This is the template for all behavior so that wolves and dogs in everyday life are always working to conform everything they do toward maintaining successive positions along a circle. What draws one’s attention, draws the others. This draws them to various objects as vehicles for triggering deep inner stress, and then resolving the subsequent pressure by recapitulating a circle.

While I can completely understand why someone would interpret this as control over access to a resource, nevertheless at some point they have to ask why doesn’t this interpretation produce a coherent model? At some point they should ask out of sheer intellectual curiosity, how might these pack interactions over low level resources be related to the simple system of hunting the highest level resource, an activity which is now demonstrated to a mainstream scientists’ satisfaction to not rely on a centralized authority or any kind of communication and rational understanding whatsoever, an interpretation which is the exact opposite of injecting a human psychology into the control-over-access-to-resources interpretation of behavior. If hunting is orchestrated by way of a flow principle rather than a centralized agent, is not social life and absolutely every interaction orchestrated by way of a flow principle rather than one agent striving to achieve control over another? If hunting is conducted without communication, then it is not likely that a concept such as “I’m in control over this resource” is being communicated when we observe social resistance between two members of a pack. If one individual doesn’t control another in the hunt, one individual isn’t trying to control another over a resource. Remember, in the hunt: “Agents do not use one-to-one communication. All communication is based on sensing the external state of nearby agents.” Hunting proceeds by referencing the external states of one’s peers so as to execute a circle, isn’t that also most likely what is being referenced in instances of social friction? In other words, there’s not enough distance between two individuals over an emotionally charged object that brings their deepest level of  constitutional tension to the surface. Since there isn’t a common object of attraction that can absorb and conduct their combined charge, we observe friction and the acquisition of stress.

As mentioned earlier, even with the prey encircled the hunt is not over. The final gap between predator and prey still has to be closed and the mathematical formula doesn’t address this. We need a singular dynamic that not only regulates attraction relative to resistance so that on the one hand it maintains discrete distances and spacing when dealing with a combative prey, but then ultimately regulates release from this pressure so that the gap is closed and the prey brought to ground. For this let us return to the notion of emotional formatting, and in particular, Deep Inner Stress.

Stress, caused by resistance to the expression of emotion, can-only-get-out-the-way-it-went-in. In other words, whatever caused resistance to the expression of emotion so that it had to be internalized and stored as stress, is thereafter necessary as a trigger. This is what I mean by formatting. Stress is acquired and layered as physical memories in the body/mind. This emotional battery then operates as a kind of tuning device, orienting and drawing the individual to specific degrees of resistance in the environment in order to find release from an ever present sense of constitutional tension. A trigger doesn’t have to be the exact same thing, to provoke it just has to be of the same intensity value as the original source of resistance that caused its formation. The Big-Brain then construes the individual’s perception of its reality so that it sees in the moment what was experienced in the past. This is why canines are drawn to tussles over resources, it’s an indirect way to actively express stored resistance, or unresolved emotion. And they next compute a circle around a common object of attraction by adopting various angles of deflection.

The specific intensity of resistance triggers a specific layer of stress. Large, dangerous prey animals trigger the deepest levels of stress, the layers that were acquired by the intense pressure brought to bear on the cubs in their earliest weeks of life, and then during “discipline” by the elders (which is really formatting and tuning) as the young approach adolescence. The cubs go too far from the den and an adult rushes over and scares them back into the “circle.” They are acquiring a deep layer of stress that will then be evoked thereafter in moose encounters. Thus we have attraction leavened by resistance when as adults, the wolves confront the large dangerous prey and specific distances are thereby evoked.

So physical memory and a formatted emotional battery executes a circle, but what then turns the pressure from something that limits to something that arouses so that the predatory aspect of the moose, bison, musk ox, elk, becomes a stimulant rather than an inhibitor? Two factors. The wolves when configured in a circle have achieved a harmonic state of alignment and synchronization. This is the same factor by which dogs shift from being car sick to starting to enjoy the ride in the car. So even though the situation is static and stressful, this emotional state  of being aligned and in sync with each other around a common object of attraction, induces a feeling of internal movement, just as if they are running at full speed after a prey that is fleeing. They can thus sustain their attraction without having to either fight or flight, which is what any other species of a lower emotional capacity would have to do. This is why aggression over “resources” (or charged objects as triggering agents) do not erupt into violence. Effecting a circle induces a feeling of movement and this softens both individuals.

Secondly, by encircling the prey they are focusing an intense emotional charge on it,  overwhelming its emotional capacity. It becomes unnerved, i.e. “ungrounded” so that it begins to physically vibrate, (rapid shallow breathing, nervous flighty mannerisms, twitchy muscles, shifty footwork, faltering gaze.) This intense vibration of the prey is the equal and opposite of the internalized deep inner stress in the emotional battery and thus releases the wolf from the inhibitory affects and repulsive properties of the deepest stress around which its body/mind has been formatted over the course of the first two years of life. It now feels an overwhelming force of attraction manifested as an uninhibited oral urge and it has thus emotionally migrated from whatever spoke on the wheel it normally occupies to the pole of Direct and Active. Because the prey is absorbing their combined emotional momentums due to its con-fusion, i.e. ungrounded-ness, each individual can occupy the Direct and Active polarity and are now free to bite as hard as they can. The pressure bubble pops as the predatory aspect of the prey is no longer an interruption, but a stimulation, the “negative” now equals access to the “positive” and the prey is brought down. The circle is complete.

At this point, all we need ask is whether there is a force already present in nature that manifests not only as a force of attraction, but simultaneously as a force of repulsion, and which can be described in precise mathematical terms, and which can also manifest as a circle generated by angles of deflection? Yes indeed there is, electromagnetism.

It is the theory of Natural Dog Training that emotion, and its equal and opposite complement of stress, working through neurology, physiology and anatomy, makes an animal feel just as if it is an electromagnetically charged particle of consciousness. I’m not saying there is a measurable electromagnetic field. I’m saying that the internal affects and mental processes induced by emotion and stress make it feel as if it is dealing with an electromagnetic field and current so that a circle will be effected through their collectivized behavior. This is the universal operating system of the animal mind and tapping into its principles of flow is how animals “know” what to do, or don’t “know.” They can either feel it, or they can’t. It’s all a function of emotional capacity and a harmonic pathway of learning. In short, Temperament is a circle.

Published March 7, 2013 by Kevin Behan
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6 responses to “The Math Underlying Natural Dog Training”

  1. wetnosewarmhearts says:

    “Math is nature’s way of communicating.” I heard this quote on a TV show re-run, NUMBERS, a few nights ago and thought of you. It seemed on target with NDT.

  2. While I do find some study of mathematics to be fascinating, I think it’s just one of the many tools that can aid us understanding the true nature of the universe. I actually find it quite odd that people are even searching for a mathematical equation to “prove” a theory of dog behavior. Isn’t that like asking for the equation of love? Or the equation of: “why do we exist in the universe”? Some things are meant to perceived, and better understood by feeling, not thought. If we had math to explain everything, life would be too easy. It would be like analyzing one’s life by saying: I made that decision in my life based on x+x squared. And then wouldn’t we be able predict the future of our lives, by calculating our present trajectory? I’m not saying there isn’t math behind everything, but really, is it necessary or even practical to use math in order to describe life in this way?

    I think once we intuit that everything in our world is just energy moving from one place to another, there’s no need to measure the behavior of our dogs in equations. We should be able “see” or “feel” the movement of their energy and then the “reason” behind why they are doing what they do would become totally clear, no math needed. Now, I don’t think that this is an easy thing to do–it’s with the same difficulty that we examine our own thoughts, behaviors and actions that we should study our dogs, be mystified, and also be grateful for even the smallest insights.

    In my opinion, Kevin, you have precisely hit the nail on the head with this statement: “…all we need ask is whether there is a force already present in nature that manifests not only as a force of attraction, but simultaneously as a force of repulsion, and which can be described in precise mathematical terms, and which can also manifest as a circle generated by angles of deflection? Yes indeed there is, electromagnetism.” And I would disagree when you negate a “measurable electromagnetic field.” Why shouldn’t there be a measurable field? (Perhaps only because our minds are not ready to perceive, believe or measure it as of yet.) My theory is that once we realize we are all just fields of energy, we won’t need math to describe our paths. We will intuitively understand the “field” that we are in, and why our energy is moving in a particular way through this field…

    Ultimately, I believe that because the beginning and end of our universe is still a mystery, then to some extent our lives, and the lives of dogs must be as well. However, if it makes people feel better to give a description to this mystery by some complicated equation that can be written down on paper, proven, taught and understood through the language of mathematics, then I guess it couldn’t hurt. Although, if all this thinking causes too much pressure in the head, then maybe it’s best to just focus on the little brain in the gut and “see” how it all feels.

  3. kbehan says:

    Very, very well said. The point with my article is to show how the latest science does in fact validate the emotional approach to understanding animals, when normally it is used to deny it. There are not two simple rules, there is but the one heart.

  4. Russell says:

    With all the descriptions very much in the frequency domain I wondered if perhaps the brain works in the frequency domain. This would certainly make a great deal of sense, temperament relates to phase, aggression – distortion/stability, emotional capacity – dynamic range, and the whole idea of time being unimaginable for a dog because they don’t have the necessary brain structure to reconstruct the memory in the time domain. This basic idea has already been proposed in holonomic/holographic brain theory. Being able to optimise behaviour, based on the brain as a frequency domain filter, would also be easier.

  5. kbehan says:

    Sounds intriguing, could you expand on the concept of a frequency domain. For example, would it be logical to speculate that each affective system that Panksepp has identified, has a frequency that we could analyze over a span of time, and that would make one concentrate on that specific system, however, they could also all be combined into a signal, a frequency domain, and this could be the underlying emotional impulse of attraction, one embedded with the values specific to each affective system?

  6. Russell says:

    The frequency domain is just another way to represent a signal, related to the time domain through the Fourier transform. I’m really not sure how holographic brain theory relates to Panksepp’s work, but I’m trying to understand some of both areas.

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