Concepts->Notes on Time
November 20, 2001
Notes on Programming Neural Systems
Measuring Information as a Function of Time
This article discusses the informational content in a sequence
of identical pulses occurring as a function of time. The smallest
observable time interval capable of measuring the pulse is computed by
iteratively determining in which half of the time interval, T, that
the pulse occurred in. Using this iterative method we can construct an
operational definition of how much information is contained in this
This procedure applies Kolmogorov's method of determining the
precision or information density in a signal to computing the information
of a sequence of pulses occurring in a time interval T.
The cornerstone of Kant's philosophy is the operational distinction
he made between the perception of reality and the the fact that
there exist an object of that perception. Kant integrated David
Hume's empiricism into his metaphysics, but he did not give up
completely "a prior" based knowledge. Kant is very hard to
understand, but he moved us in the direction of seriously trying
to understand and study how we perceive objects.
As we study how we perceive things, we learn that the underlying
reality lies not only in our perception of objects, but that
reality is actually wrapped up in the mechanism of perception.
From experimental physics, we know that we can only observe or measure
an event in time to an accuracy limited by the uncertainty principle, ie.,
dE * dt >= h,
where dE is the energy of a signal, dt is the time duration of
a signal and h is Planck's constant. A detector cannot measure an
event in time with accuracy greater than dt where
dt >= h/dE.
Note, that the limit of temporal resolution separating two time
dependent Gaussian pulses is the standard deviation (one-half the width
at the half-height of the pulse). The temporal resolution, dt, is
also related to the highest physical frequency component of the signal.
Nyquist's theorem says that in order to resolve a signal of frequency f
the "carrier" frequency needs to be 2f. The temporal resolution dt
is equal to the carrier frequency of the signal 1/2f.
Physical Measurement of Events in Time
Generally speaking, let a signal be defined by the occurrence of some
physical phenomena like the observation of a sound pulse or
electromagnetic wave packet. The recording as a function of time of
this signal starts with the observation of state changes in the signal
detector like the change of amplitude or frequency. The end of the
signal in time occurs when the detector no longer recognizes any
changes in the observation of this signal.
Assume that the detector uses a computer's clock with the lowest
resolution of time defined by the CPU's single clock cycle or one
CPU clock tick. Assume too that we are measuring time ordered
sequences of identical pulses. To accurately measure a single pulse,
the clock tick, using Nyquist's Theorem, needs to be shorter then
the standard deviation of the pulse width.
Note, furthermore, that the smallest physcial duration of the clock tick
is limited by the uncertainty principle.
Information in a Time Interval
Let the duration of observing a pulsed signal be some arbitrary time
interval T. Also assume that the occurrence of pulses is random as
in a Poisson process. In the following argument, we assume that
the probability of the occurrance of the event after we measured it
is 1. The information in the occurrance of an event anywhere inside
the time interval T is 1 bit. The key idea is that we get more
information when we know where inside this interval, T, the event
0 | T
Let the informational content of this observation, that is, the
knowledge that a pulse was observed in time T be I_o. Now split the
the time interval in half, and select the half in which this pulse
resides. The informational content of this observation has been
doubled. The observable time interval has been cut in half. The
information gained is 1 bit. Repeating this process gains you
another bit of information, and cuts the original time interval
T by a quarter.
I = 2 * I_o, T -> T/2
If we repeat this procedure until we reach the limit of observation
in time, then, the informational content is
I = 2M * I_o, T -> (T/(2*m))
where M is the number divisions or splits in the time interval T.
The observable time interval is T/2M. The number of observable time
intervals (ticks) in a time ordered sequence follows Nyquist's theorem
of doubling the sampling interval per number of pulses.
Assuming the informational content is additive for each pulse, the
information content in a sequence of N pulses in the time interval T
I(T) -> N * 2M * I_o.
If you have N pulses, you have to have 2*M * N time intervals in
order to be able to assign a value for each pulse.
The maximum information content of the signal I occurs when you
make T equal the difference between the first occurrance and last
occurrance of the sequence of pulses.
0 | T
T = tF - t0
2M * t = T or 2M = T/t
f = 1/t
2M = Tf
I = (N*T) f
The informational content of the signal in the interval [0,T] is
proportional to the number of pulses occurring in this interval,
and inversely proportional to the observation time of the shortest
temporal duration t. This dimensionaless number for I is a measure
of the informational content in a time interval obtained by subdividing
the total observation time of the signal in a binary partition. The
unit for measuring this information is obtained by reducing the
observation time down to the smallest operational temporal unit called
the "binary interval." I stress the word interval for its sense of
If you, somewhat arbitrarily, standardize the observable time unit T
making it 1 second, then I = N * f, and we could call this unit of
measure the bint. The bint would equal the number of pulses
occurring in one second measured with a large enough frequency of
observation to detect the pulses. The bint is equivalent to the
hertz as a measure of energy. The hertz could also be use as an
explicit measure of dynamic information in the right context.
Information Transfer Rates
Nyquist's rule for the minimum information content required for the
observation of a single pulse is 2f. Information rate is related
to number of pulses in the signal in time T or the intrinsic binary
interval dt. (~= means proportional to)
R_max ~= log (2 N); N = number of pulses in the time
R_max ~= log (2/dt); 1/dt describes to how much pulses
can be placed in a unit time
Measuring the Instrinic dt in Neurons
We may be able to find the physical smallest binary interval length used by
neurons by examining the uncertainty principle dt = h/dE. An
example of nature building a biological organ to the limitations
of a physical system is the human eyes. Human eyes have evolved to
resolve images up to the physical optical resolution limit. Similarily,
the lower limit on pulses triggered by neurons only exist for the duration
limited by dt which is define in the uncertainty principle: dt >= h/dE.
1. "Entropy and Codes" by Fleming Topsoe, http://www.math.ku.dk/~topsoe/
This article uses Kolmogorov's method of obtaining information content
to define information in binary codes.