**Name:**

growth_curve_gaussian - Gaussian version of a growth curve

**Description:**

This class represents a Gaussian growth rule for the number of synaptic

elements inside a neuron. The creation and deletion of synaptic elements

when structural plasticity is enabled, allows the dynamic rewiring of the

network during the simulation.

This type of growth curve uses a forward Euler integration method to update

the number of synaptic elements:

dz/dt = nu (2 * e^(- ((Ca(t) - xi)/z)^2 ) - 1)

where xi = (eta + eps)/2,

zeta = (eps - eta)/2 * sqrt(ln(2))),

eta is the minimum calcium concentration required for any synaptic element

to be created, eps is the target mean calcium concentration in the neuron

and nu is the growth rate in elements/ms. The growth rate nu is defined in

the SynapticElement class.

**Parameters:**

eta double - Minimum amount of calcium concentration that the

neuron needs to start creating synaptic elements.

eta can have a negative value, making the growth

curve move its maximum to the left. For example, if

eta=-0.5 and eps=0.5 [Ca2+], the maximum growth rate

(elements/ms) will be achieved at 0.0 [Ca2+]. If

eta=0.0 [Ca2+] and eps=0.5 [Ca2+] the maximum growth

rate will be achieved at 0.25 [Ca2+] while at 0.0

[Ca+2] no new elements will be created.

eps double - The target calcium concentration that

the neuron should look to achieve by creating or

deleting synaptic elements. It should always be a

positive value. It is important to note that the

calcium concentration is linearly proportional to the

firing rate. This is because dCa/dt = - Ca(t)/tau_Ca

+ beta_Ca if the neuron fires and dCa/dt = -

Ca(t)/tau_Ca otherwise, where tau_Ca is the calcium

concentration decay constant and beta_Ca is the

calcium intake constant (see SynapticElement class).

This means that eps can also be seen as the desired

firing rate that the neuron should achieve. For

example, an eps = 0.05 [Ca2+] with tau_Ca = 10000.0

and beta_Ca = 0.001 for a synaptic element means a

desired firing rate of 5Hz.

nu double - Growth rate in elements/ms. The growth rate nu is

defined in the SynapticElement class. Can be negative.

**References:**

[1] Butz, Markus, Florentin Wörgötter, and Arjen van Ooyen.

"Activity-dependent structural plasticity." Brain research reviews 60.2

(2009): 287-305.

[2] Butz, Markus, and Arjen van Ooyen. "A simple rule for dendritic spine

and axonal bouton formation can account for cortical reorganization after

focal retinal lesions." PLoS Comput Biol 9.10 (2013): e1003259.

**Author:**

Mikael Naveau

**FirstVersion:**

July 2013

**SeeAlso:**

**Source:**

/home/graber/work-nest/nest-git/nest-simulator/nestkernel/growth_curve.h