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This site exists to explore a linear model of the human  brain.  The brain is supposedly not linear at the microscopic level, but it can be argued that in a global sense it is a linear system.

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In the social sciences, aspects of human personality and behaviour are studied using linear algebra.  The use of that as a tool does not imply that the human brain is linear, but its considerable utility is highly suggestive.  If the human brain does turn out to be linear, then the use of factor analysis and other methods of linear algebra in the social sciences would be validated.

A linear system is one that obeys the principle of superposition, which states that the response to two stimuli presented together is the sum of the responses to each stimulus alone.  Since multiplication can be defined in terms of addition, this additive property leads to the further homogeneity property that if the stimulus is multiplied by some factor, the response is multiplied by the same factor.

Linear systems are studied using linear algebra, and represented by differential equations of the first order, though not all differential equations of the first order are linear.

One concise way of describing a linear system is that its outputs are proportional to its inputs.  I argue that this applies to the human brain.

I can think of many ways in which this just seems to be false.

But there are at least two good reasons for thinking it’s true .  One is empirical, based on an examination of the evidence, while the other is evolutionary, based on the probable survival value of the linearity property.    Both are presented here on this website.

A related website is that for linear profiles.  Those may be created and used even if the human brain is not a linear system, but evidence that it is would also be evidence that they have a mathematical validity.

What is the empirical evidence?  It has to do with multi-tasking.  We are not digital systems like computers which can slice up their processing time and seem to do many things at once, while in fact following a single sequence of steps.  We are instead analogue devices, despite the apparent all-or-nothing response of individual neurons.

It must be noted that while it is impossible to build a non-linear system from linear components, it is possible to build a linear system from non-linear components, since the non-linearities can cancel each other out.  It is argued here that this must be so for the human brain, since there is clear evidence that we are linear systems.

In such a system, the internal communications paths cannot carry multiple signals without some crosstalk — an interference between the various signals.

While we do sometimes show a small amount of this interference, the extraordinary amount of communication and control taking place during human activities could not take place if such crosstalk was significant.

Therefore, the human brain must be essentially linear in the large.  How is this possible, when there are so many apparent non-linearities at the microscopic level?  That is one subject for discussion on this website.

There are at least two approaches to explaining how the human brain could be linear.

  • One, already mentioned, is that non-linearities can can cancel each other out.  While a possible explanation, it is not very specific
  • Another is to deny that the brain is non-linear at the microscopic level.  If the important variable is not the strength of the electrical charge on a neuron but the timing of its changes, then the all-or-nothing rapid pulses of a neuron firing do not by themselves represent a non-linearity.

It seems very possible that the rate in which a neuron fires is proportional to the concentrations of certain chemicals in the vicinity of the synapses.  These may in turn be proportional the rate at which connected neurons fire.  So it may all be a matter of timing, and in that sense completely linear.  This is not proven, but is a plausible model of how the brain works, consistent with the empirical evidence.

Both of these explanations could be true.  Some non-linearity in the timing of neural events may be cancelled out by others in the brain.

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