What is thinking?
The dynamics of mental exploration
‘Thinking’ is a process by which a
computational system can generate an
effective action in a novel situation, based
on exploring the possibilities (often
combinatorial) implicit in previously
acquired knowledge.
Sound waves in a simple gas
molecular viewpoint
m dvj/dt = Fj
molecules j of mass m, molecular forces
(Newton’s laws for 1028 molecules)
Fj = - Sk grad V(rj – rk)
no sound waves without molecular interactions
Navier-Stokes viewpoint
pressure fluctuations in a fluid: mean density r compressibility k
k-1 (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2) p = r ∂2p/∂t2
The existence of molecules and their microscopic
interactions has disappeared
Drift and diffusion model of decision-making
Sliding motion along a coordinate
Drift caused by available evidence (slope)
Diffusion (random walk addition) caused by noise
Decision made when green dot reaches an end
Reaching lower end means decision correct
Reaching upper end means erroneous decision
Mental exploration
A protracted evolution of the pattern of neural activity,
while an animal is not yet taking actions and
while sensory input may be constant (or irrelevant),
followed by an apt behavioral action
that directly relates to the activity states during the exploration.
Box environment with
visual clues on walls
Where action potentials ‘spikes’
are generated in a typical rat
hippocampal place cell during
exploration of a familiar space
from Wills, Lever, Cacucci,
Burgess and O’Keefe (2005)
Science 308, 873
Mapping brain cell activity to a useful spatial display
cell location in brain
spatial display for a rectangular environment
arrowheads at position of maximal activity
of corresponding neuron
‘’near water” neuron
(connections learned)
Spatial representation
of activity pattern
(reordering of pattern
in hippocampus)
Strongly active neurons
in hippocampus
when animal is at
location w
o
from Wills, Lever, Cacucci,
Burgess and O’Keefe (2005)
Science 308, 873
Animal in
R environment
Animal in
H environment
Rectangle
environment
display
H environment
display
3
3
0
0
-3
-3
0
3
-3
-3
Mental exploration
While stationary, mentally explore extensively to search for water in
present environment
make a stable activity clump in any particular environment
make the activity clump explore that environment
If water is found, find a (the?) mental pathway between present physical
location and location of water
Remember the pathway so that it can be mentally repeated
Use mental recapitulation of motion to guide corresponding physical
motion along the mental path
dik/dt = -ik/syn + S Skj f(ij + Isensory )
f(x) something like 0.5*(1+tanh(x))
Skj = Tkj - 
Tkj represents specific excitatory synapse from neuron j to k
 term is all-to all global inhibition
If T symmetric
notation fk = f(ik)
Lyapunov function
L = 1/syn
f
S ∫ k f-1(x)dx
- ½ SS Skj fjfk
Exploration in one dimension
Each neuron has a location of maximum activity
Arrange neurons in this natural order for display purposes
Connect each neuron k to M others (denoted by j) that are most
strongly active when k is active.
This defines the connection matrix Tkj
“neurons that fire together wire together”
c
Tkj
3
3
0
0
-3
-3
0
3
-3
-3
input
Firing rate adapation
dik/dt = -ik/syn + S Skj f(ij + Isensory + aj)
dak/dt = -ak/undadapt -  f(ik + Isensory + ak)
Skj = Tkj - 
If syn = unadapt, change of variables to xk = ik + ak
dxk/dt = -xk/syn + S Rkj f(xj + Isensory )
Rkj = Tkj -   kj
would have a Lyapunov function
but syn << unadapt
Adaptation produces a qualitatively new behavior
yielding
Mental exploration
While stationary, mentally explore extensively to search for water in
present environment
If water is found, find a (the?) mental pathway between present physical
location and location of water
Remember the pathway so that it can be mentally repeated
Use mental recapitulation of motion to guide corresponding physical
motion along the mental path
Mentally learning a physical trajectory
move along a path
sensory input dominates place cells
Skj = (activity of neuron k )* (activity of neuron j)
accumulate Skj throughout this motion
For all synapses that are non-zero (i.e. Tkj = 1)
if Skj > threshold value
increase Tkj by 50%
{Strengthens synapses that would be useful along the path}
MLS
Motor controller
integrate-and-fire neuron
slow excitatory pathway
each output spike
approximately reverses
direction of motion with
random spread  60o
input
balanced fast inhibition
long-lasting
self-inhibition
[Ca-dependent
Inhibitory currents]
No control signal
control input from olfactory cells
Gaussian spatial odor profile
3
3
0
0
-3
-3
0
3
-3
-3
0
3
MLS
Area E activity is a moving bump representing intended action
Area A has two ‘bumps’ of input
Sensory input represents where the animal is
Input from area E reflects the intended position
When these coincide, area A has maximal activity
When well separated, area a has little activity
Motor system will move animal to intended location
mentally explore for w
recapitulate mental success to
guide physical motion
Activity-position movie
10 frames/sec
Red * instantaneous location of animal
Black points .
Center of receptive field of strongly active place cells
100 randomly chosen place cells in the interior of a T environment
Explanation of movie
The mouse goes one branch to another, not directly repeating.
(perhaps a learned behavior)
Hippocampal place cells have indirect inputs from the sensory+vestibular system.
They have a selective filter on this input resulting in spatial receptive fields.
When the animal is at X, the cells with place field centers near X
are strongly active, driven by sensory input that characterizes being at X
{Being at X causes corresponding neural activity WRONG}
There is NO sensory input to these place cells (in E = ca3?) during the movie.
A moving cluster of place cells with is active through mutual feedback.
When this cluster is at X, the animal ‘wants to be at X’
(i.e., this activity causes the motor system to move the animal to X).
Intrinsic neurodynamics makes the active cluster moves in mental space,
causing the animal to move correspondingly in real space.
CAUSALITY IS REVERSED
dik/dt = -ik/syn + S Skj f(ij + Isensory + aj)
dak/dt = -ak/undadapt -  f(ik + Isensory + ak)
Skj = Tkj - 
syn << unadapt
Adaptation produces a qualitatively new behavior
With appropriate (learnable) synaptic connections S this high
dimensional system (>10000) has a dynamical behavior in which
there is rapid relaxation to a slow manifold of low dimensionality
(eg 2 ), the space on which a collective motion occurs
A statistical physics problem lies at the heart of mental exploration
and the relationship between neurobiology and psychology
Given S, what are the collective variables and their
equations of motion?