604 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 10, NO. 3, MAY 1999

Finding the Shortest Path in the Shortest Time Using PCNN’sH. John Caulfield and Jason M. Kinser

Abstract—A pulse coupled neural network (PCNN) can runmazes nondeterministically (taking all possible paths) with con-stant time per step. Thus, when a signal emerges, it has taken theshortest path in the shortest time.

Index Terms— Autowave, maze, nondeterministic solutions,pulse coupled neural networks, shortest path.

I. INTRODUCTION

T HE subject is maze running, but the applications are verybroad since paths and path lengths may represent almost

any combinatorial components.Our objectives are twofold: find the shortest path and do

so with the minimum effort. By inspection, no algorithm cando both. How could we guarantee that one path is optimumwithout at least having examined other paths? Yet, clearly, theminimum effort approach is “take the shortest path.” Theseconsiderations point to a nonalgorithmic, “nondeterministic”approach.

One way of explaining the nondeterministic approach is thisone. We have a semiinfinite number of idiots each trying thingsand keeping a record of what they tried. Nearly all will fail tosolve the problem, but (if we are lucky) one will succeed. Thatis, we simply try everything at once and crown the winner. Wethen ask, “how did he do it?”

II. BASIC APPROACH AND EXAMPLE

Our approach is to use a pulse-coupled neural network(PCNN) as pioneered by Johnson [2]–[4]. This is a biologicallybased system [1], [7], [8] shown symbolically (for a singleneuron) in Fig. 1. Details are available in the references andelsewhere in this issue and omitted here for brevity.

In all prior PCNN’s, the input has been a full array of imagepixels in a steady state. Here we introduce departures fromthat case. The input and output pixels are no longer an evenlyspaced array. Rather, they are points in the maze. For example,see Fig. 2.

The PCNN is an iterative procedure in which the outputfrom one iteration stimulates the next iteration. The input tothe maze is constructed so that the paths are one pixel in widthand have the same intensity. The initial starting point (labeled“start” in Fig. 2) is different in that its intensity is considerablehigher than the other maze points. In the initial iteration only

Manuscript received August 8, 1997; revised November 1, 1998. This workwas supported by the U.S. Army Space and Strategic Defense Command underContract DASG60-95-2-0001 and the U.S. Army Missile Defense Commandunder Contract DAAH01-93-X-R351 while both authors were at AlabamaA & M University.

H. J. Caulfield is with the Fisk University, Nashville, TN 37208-3051 USA.J. M. Kinser is with the Institute for Biosciences, Bioinformatics, and

Biotechnology, George Mason University, Manassas, VA 20110 USA.Publisher Item Identifier S 1045-9227(99)03928-4.

Fig. 1. The schematic of a single PCNN neuron.

Fig. 2. The example maze. The task is to procedure along a path from “start”to “end.”

the “start” pixel has sufficient intensity to pulse the neuron.The dynamic threshold is set to a positive value that insuresthat only the “start” pixel has sufficient internal potential topulse.

The input stimulus is then turned off for all subsequentiterations. In the second iteration the one pixel that pulsedcreates an autowave [6] that can only travel through thoseelements which have a nonzero internal state (i.e., thoseneurons which had an input stimulus from the first iteration).Once a neuron pulses the dynamic threshold is increased toa very large value that will prevent this neuron from firingduring the rest of the computation.

The autowave will then travel forward along the mazesimultaneously through all possible paths.

The output of each iteration is accumulated with a decayterm. Therefore, in this accumulation the most intense pixelswere the ones that pulsed most recently. The shortest pathwill be the path whose pixels in the accumulated output aremonotonically increasing in intensity from “start” to “end.”Fig. 3 displays this accumulated pulse field after selectediteration intervals. The parallel progression of the tracks isobvious. The PCNN stops when the “end” pixel pulses. Thenumber of iterations (and equivalently the number of steps orcomputation cycles) is proportional to the length of the shortestpath to the output. The complexity of the maze, the number

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IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 10, NO. 3, MAY 1999 605

Fig. 3. Selected frames of the weighted average accumulation pulse field.

of alternate paths and intersections, does not alter the numberof computations.

Determining which path is the correct path is quite easyand a computation that is still dependent upon the path lengthand the number of intersections along that path is all that isrequired. Since the final output is a time weighted average ofthe values, the elements of the correct path are monotonicallyincreasing from “start” to “end.” Starting from “end” theshortest path is determined by back tracing along the onlymonotonically decreasing path.

Confusion can not occur at the junction points during thistrace back. When an autowave comes upon a junction point,the wave takes all possible forward paths with the same wavevelocity. Thus, all exiting paths have the same increasingvalues in the accumulated output field and the single inputpath is quite distinct from the exiting output paths. A searchfor the correct path only needs to look at two pixels alongeach possible path to determine which was the incoming path.During trace back, this is the only type of path being sought.Fig. 4 shows the shortest path for this example.

As stated earlier, the number of computations in the traceback is weakly dependent upon the complexity of the maze.Basically, this dependence is dependent upon the length of thepath and the number of junctions along this path.

III. D ISCUSSION

It should be noted that the original maze had some otherspaths that connected “start” to “end.” The trace back in Fig. 4

Fig. 4. The result of the trace back.

displays the shortest path. The other two paths enter at thetop of the circle in the maze, arc about either side, and thenconnect to “end.”

In a sequential digital computer simulation, the complexityof each step depends solely on the number of neurons. In a par-allel computer, the amount of computation time could dependsolely on the time to perform a single neuron computation(given one processor per neuron). The fast way to run a bigmaze is to use an optical PCNN [2] which can, with currentoff the shelf technology, operate a 10241024 neuron arrayat 1000 Hz frame rates. It should be possible to construct a

606 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 10, NO. 3, MAY 1999

complex maze branch in a 10 10 neuron or pixel array,so we can imagine up to 10such branch points. Thus, weshould be able to solve large NPC problems quickly using anoptical PCNN.

An electronic parallel implementation of a PCNN has beenaccomplished on a CNAP’s computer [5]. This system was aSIMD architecture with 128 processing elements and achieveda speed of about 30 PCNN iterations per second.

Philosophically oriented readers may enjoy the observationthat one way to describe quantum mechanics (mostly associ-ated with R. Feynman) is identical to this. The realization thatlight takes the shortest path is at least 2500 years old—Heroof Alexandria. Feynman pointed out that an infinity of pathsmust be explored to accomplish this, which is rather a complexcomputation achieved in a finite time. He suggested that acomplex wave travels all paths. The resultant net complexwave patterns cancel one another out leaving only the shortestpath. Thus, nature too uses a nondeterministic, nonalgorithmic,explore-all-paths approach to finding the shortest path in theshortest time. Thus, the PCNN implements a crude “artificialphysics.”

REFERENCES

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[2] J. L. Johnson and D. Ritter, “Observation of periodic waves in a pulse-coupled neural network,”Opt. Lett., vol. 18, no. 15, pp. 1253–1255,1993.

[3] J. L. Johnson, “Pulse-coupled neural nets: Translation, rotation, scale,distortion and intensity signal invariances for images,”Appl. Opt., vol.33, no. 26, pp. 6239–6253, 1994.

[4] J. M. Kinser and J. L. Johnson, “Stabilized input with a feedback pulse-coupled neural network,”Opt. Eng., vol. 35, no. 8, pp. 2158–2161,1996.

[5] J. M. Kinser and Th. Lindblad, “Implementation of the pulse-coupledneural network in a CNAPS environment,” accepted by this issue, pp.584–590.

[6] O. A. Mornev, “Elements of the ‘Optics’ of autowaves,” inSelf-Organization Autowaves and Structures far from Equilibrium, V. I.Krirsky, Ed. New York: Springer-Verlag, 1984, pp. 111–118.

[7] I. A. Rybak, N. A. Shevtsova, L. N. Podladchkova, and A. V. Golovan,“A visual cortex domain model and its use for visual informationprocessing,”Neural Networks, vol. 4, pp. 3–13, 1991.

[8] H. Sompolinsky, D. Golomb, and D. Kleinfeld, “Global processing ofvisual stimuli in a neural network of coupled oscillators,”Proc. Nat.Academy Sci. USA, vol. 87, pp. 7200–7204, 1990.