NPB 163/PSC 128

Dynamics

Differential equations

• Dynamics is essentially the study of how things change over time. This is important for understanding the brain, because we are constantly being inundated with time-varying signals. How neurons respond to these signals over time is the essence of what neural coding is all about.
  
  
• Differential equations simply provide a mathematical description for how things change over time. The central element of a differential equation is the time-derivative of a variable, . This essentially tells us the rate of change of the variable x, where x(t) may be voltage, position of a particle, etc.
  
  
• The derivative can have different orders: , etc. For example, if x is position, then will be its velocity and its acceleration.
  
  
• A first-order differential equation would have terms of the form x and . A second-order differential equation would have terms of the form x, , and . Or in general, an -order differential equation would have terms of the form x, ,... .
  
  
• The simplest kind of differential equation is a linear differential equation, which simply contains derivatives of different orders, each multiplied by a constant:
  
                   
  
  
• Oftentimes, we denote a time-derivative using the “dot” notation . Thus, in this notation, a second-order linear differential equation would have the form
  
                     
  

• The very simplest differential equation would be a first-order, linear differential equation:
  
                     
  
  
• When , then we have the simple relation . What this tells us is that the rate of change of x depends on the value of x. If x is currently a large positive value, then x will decrease quickly. If x is currently a negative value, then x will increase, etc. Is there a mathematical equation that will tell us explicitly how x changes as a function of time?
  
  
• It turns out that the general solution to the above equation is of the form
  
                   
  
  where the constant k is determined by the initial condition, or the initial state of x at t=0, and . For example, if x(0)=1 and , then we have . Thus, a first-order linear differential equation describes the process of exponential decay.
  
  
• The rate of decay is determined by the “time constant,” . If is large, then this means that x decays slowly. If is very small, then x decays quickly. Basically, the way to think of it is that when an amount of time has gone by, the value of x will have been reduced by a factor of 1/e (the number e is about 2.7).
  
  

• So far, we have examined the case where the right-hand side of the differential equation is zero. What if there is a time-varying function on the right hand side? i.e.,
  
                     
  
  How does vary as a function of ?
  
  
• It turns out that the solution for is now
  
                     
  
  where the function is the solution obtained when the right-hand side is zero. Thus, this equation tells us that is simply a linearly weighted sum of the present and past values of . The weights are given by the function , which as we have seen above is an exponentially decaying function. So the more recent values of will be weighted more heavily than the present values, and values far in the past will be entirely forgotten.
  
  
• Such a system is called a leaky integrator, because the past leaks away from the summation of present and past values of . We can think of as the input to the system and as the output of the system
  
                   
  
  where characterizes, via its time-constant, how far in the past that values of will affect the current value of . Thus, acts as a filter on the function , smoothing over its details. If is large, then smoothing will be severe. If is small, little smoothing will occur.