Echotrigger/Scopion Theory
of the Function of Neurons in the Imaging Sonar of Dolphins 
More Details Here

There is a feature of the clicks of dolphins that I call a “fang.” An instance of a fang is a change in waterpressure from low to high to low occurring in about 1/100,000 second that involves a much greater such change than any other in the click. A click has cycles of "ringing" that follow its fang. 
FIG 1 Fang in click 
An “echotrigger” is an hypothesized kind of sensor that sends a pulse along an axon into the brain when a fang of a click passes across that sensor. 
A ditoaneuron has exactly two input dendrites and one output axon and is such that the output fires only after the nearlyenough simultaneous firing of the two inputs. "Ditoa" comes from DifferenceInTimeOfArrival. FIG 2 shows an icon that represents a ditoaneuron. 
FIG 2 DitoaNeuron 
FIG 3A 
Suppose there are two echotriggers M1 and M2 on the right and the left side of the head, respectively.
Relating to FIG 3A, suppose that a ditoaneuron N, on the left side of the brain, has an input that is a branch of an axon from M1, and the timeoftravel, of a pulse from M1 to N, is s1. Likewise, suppose N can receive an input from M2 after a timedelay of s2 sec. Since the path from M2 to N is shorter than the path from M1 to N, s2 < s1. 
A soundreflecting point P might be positioned to the rightfront of the head so that, when an echo of a fang leaves P, it travels to M1 in t1 sec and to M2 in t2 sec such that t1+s1 is so nearly enough equal to t2+s2 that the ditoaneuron N fires. 
FIG 3B 
FIG 4 The set of all points P, that can fire ditoaneuron N, lies on an hyperbolic surface we can denote by H(M1,M2,N). In reality, given the fuzziness of Nature, it is a fat hyperbola, not just a single point thick. 
In Fig 5, M1 and M2 represent acoustic sensors on the face of a dolphin. The color of a dot P in those horizontal rows in space in front of the dolphin is the color of a ditoaneuron in DitoaArc(M1,M2), seen in FIG 5, that would fire were a click to emanate from P.
If a wave leaves the indicated point P, it reaches M1 along the straight, blue path sooner than it reaches M2 along the red path. Neuronal signals, triggered by the wave, and leaving M1 and M2, arrive nearly enough simultaneously at the bold green ditoaneuron to cause it to fire. The DitoaArc(M1,M2) determines direction to sound in the leftright sense.

FIG 5 
Now, consider some geometry and arithmetic: Imagine there is just one soundreflecting point P and it is located to the right front of the head of our imagined dolphin. Let t1 and t2 denote the timeoftravel of a fang from P to M1 and to M2, respectively. Since P is nearer to M1, t1 < t2. Suppose that the ditoaneuron N is on the left side of the brain and has an input that is a branch of an axon from M1, and that s1 denotes the timeoftravel, of an axonal pulse from M1 to N. Likewise, suppose N can receive an input from M2 after a timedelay of s2. Since the path from M2 to N is shorter than the path from M1 to N, s2 < s1. Note the possibility that t1+s1 = t2+s2. If a fang leaves P at time T0, it arrives at M1 at time T0+t1 and generates a pulse at M1 that arrives at ditoaneuron N after an extra duration s1. N receives that pulse at time T0 + (t1+s1). Likewise, N receives via M2 a pulse at time T0 + (t2+s2). Let T1 and T2 denote, respectively, the toas T0 + t1 and T0 + t2, the timesofarrival of the fang from P at sensors M1 and M2 respectively. If a pulse from echotrigger M1 and a pulse from M2 arrive at ditoaneuron N at nearly enough the same instant, then N is stimulated to fire, sending a pulse to another level in the brain. When, nearly enough, the difference (T2+s2)  (T1+s1) equals zero, N fires. So, when the ditoa T2  T1 equals, nearly enough, s1  s2, the ditoaneuron N fires. The difference s1  s2 is internal to the dolphin and T2  T1 is imposed on the dolphin by the environment. Let the distance from a point A of 3Space to a point B of 3Space be denoted by d(A,B). The set of all possible reflecting points P that could produce exactly the same t1,s1,T1 and t2,s2, and T2 constitute a subset of a surface H12 determined by distances d(M1,P) and d(M2,P) and a number u computable from the times and the speed of sound in the medium. H12 is the set of all points Q such that
Let Speed denote the speed of waves in the medium. It can be shown that u = Speed * (s1  s2). H12 can be called a “hardwired hyperbola” of the system. It depends upon the locations of M1 and M2 and upon the axonal lengths that determine the timedelays s1 and s2. When externally impinging T1 and T2 are such that the ditoa T2  T1 is near enough to s1  s2, the ditoaneuron N fires. N would be just one of a possibly linear array of ditoaneurons that receive input from M1 and M2. Call that array DitoaLine(M1,M2). Which of the ditoaneurons in DitoaLine(M1,M2) fires as a result of a fang from a point P could signal where P is located in a lefttoright sense. If u=0 then a ditoaneuron fires that indicates P is straight ahead. As u is more negative a neuron fires that indicate further P is to the left of straight ahead. As u is more positive, the further P is to the right. It was to simplify discussion that we imagined echotriggers M1 and M2 to be equally spaced about the foreaft axis of the head. Similar reasoning would apply to any two sensors. Our instinct might be to separate M1 and M2 as much as possible, but, let us imagine that M1 and M2 are together, right and left, on the dolphin’s chin and that M3 is a third echotrigger on the chin, positioned so that Angle (M2,M1,M3) is right. The DitoaLine(M1,M3) determines direction in the updown sense. Together, DitoaLine(M1,M2) and DitoaLine(M1,M3) could determine, at least roughly, a direction from the head to a reflecting point P. Another, planar, set of neurons G could each receive input an input from DitoaLine(M1,M2) and an input from a ditoaneuron in DitoaLine(M1,M3). G could represent, to some deeper part of the brain, the direction to P, the source of the fang. We see here indications of methods available to, but not necessarily really functioning in, Nature. 
Can the imaging sonar of dolphins be explained as resulting largely from just echotriggers and ditoaneurons?
See the warnings given in:
to be continued ... 