Cell Language Theory, The: Connecting Mind And Matter

The Universality of the Planckian Distribution Equation 357

“6x9” b2861 The Cell Language Theory: Connecting Mind and Matter

Figure 8.7 The mathematical relation between the Gaussian and Planckian distributions

entailed by the Drift Diffusion Model (DDM) of decision-making. (a) The DDM adopted

from Figure 3 in [346]. (b) The experimentally observed decision-time histogram contains

a Gaussian component. (c) The Planckian distribution can be derived from a Gaussian

distribution following a simple algorithm described in the text above.

(a)

(b)

(c)

1800

decision time (t)

O

tan α = D/t = drift rate

D = decision

threshold

v(t), decision variable y =

The Gaussian component of the decision-time distribution

Data from G. Deco, E. T. Rolls, et al. (2013). Progr. Neurobiol. 103 :194-213;

Fig. 2b. μ = 120, σ = 50. S. Ji, 5/8/2014

1600

1400

1200

1000

800

600

400

200

–200

70

60

50

40

30

20

10

0

020406080

0

0 200 400 600

Decision times, ms

# of

Trials

# of

Trials

800

The Gaussian distribution transformed by the rule x=> D/tan x

becomes

a Planckian distribution!

1000

A = 32, B = –1, C = 40, RMAD = 56.676, by trial & error fitting S. Ji, 6/13/2014

x = > D/tan x,

S = Decision threshold, and x is the arc tan of drift rate μ. i.e., D/decision time.

A

(x + B)5

1

ec/(x + B)–1

e–

(x - μ)^2

y =^12 σ^2

σ √ 2 π

α

Experimental

Gaussian

x-transformed

Gaussian

Planckian

b2861_Ch-08.indd 357 17-10-2017 12:09:16 PM