Following Muller et al.~(1996), 
here we try a navigation using a cognitive map created in a recurrent connections of CA3 pyramidal cells as place cells with funtions of long-term potentiation. modeled by spiking neurons with functions long-term potentiation.

This model is based on the finding by O'Keefe and Dostrovsky (1971) that firings of fippocampal neurons in freely moving rats is location specific, that is thye fire rapidly only when the rat is in a specific lacation. So cuch neurons are now called place cells and are pyramidal cells of the CA3 and CA1 regions of the hippocampus.

<Eucledean>

Here assumption is, the mapping information, namely distance relation of the points in the environment, is represented as the strengh of long-term potentiation modifiable Hebbian synapses. In other words the mapping information is stored in the strength of the connection. Here we models it in the strengths of CA3 to CA3 synapses of their recurent connection. So, the short intervals between pre- and postsynaptic spikes are expected to cause increased synaptic strength.


Since each cell is a place cell, any path in the graph corresponds to a path in 2-D space.

the further the distance in the environment the smaller the strength between corresponding two cells.

Then the question is, the optimal paths in nerual space are optimal too in geometrical space of surroundings. This is what we show in this section.



What Muller proposed is strength of a synapse is determined according to a decreasing function of the distance between two points the two neuron represent. That is, the longer the distance the weaker the strength assuming no barrier between the two points. (if there is a barrier
it is set how far the rat must go with a minimum route to get from one point to the other.)
In other words, synaptic strength should decrease with distance between two points.


(1)
- create N cells
     every cell should have same number of n outgoing edge (only one connection to one other neuron, no me to me)

- test for strong connectivity
   it should be possible to walk from any node to any other node
(A surprisingly low divergence is necessary to virtually ensure that the network is strongly connected)

each cell is assingned a location in 2-D space at rondom real field is represented by pixcels

then synapse is given a strength according to the distance between the field centers


* Finding optimal paths in synaptic resistance space and 2-D space.