Table 3. The best optimal theoretical fractional volumes and related parameters in the cortex for “spine economical maximization” principle.

The optimal fractions correspond to either the minimal Euclidean distance (ED) or Mahalanobis distance (MD) between theory and data (given in bold face).

Spine size distribution	θ	Optimal parameters								ED	MD
		x	y	s	g	c	$\overline{u}$	P	γ 2
Exponential	0.100	0.374	0.374	0.119	0.118	0.014	0.615	0.850	0.25	0.043	1.982
	0.100	0.374	0.374	0.119	0.118	0.014	0.615	0.850	0.25	0.043	1.982
	0.321	0.398	0.398	0.093	0.102	0.009	0.599	0.585	0.50	0.050	5.913
	0.321	0.397	0.397	0.098	0.097	0.010	0.678	0.623	0.45	0.051	5.883
Gamma (n = 1)	0.100	0.366	0.366	0.129	0.123	0.016	0.626	0.959	0.20	0.051	2.353
	0.100	0.366	0.366	0.129	0.123	0.016	0.626	0.959	0.20	0.051	2.353
	0.321	0.388	0.388	0.098	0.116	0.011	0.520	0.650	0.60	0.039	3.886
	0.321	0.385	0.385	0.111	0.107	0.012	0.660	0.746	0.45	0.042	3.597
Gamma (n = 2)	0.100	0.370	0.370	0.136	0.110	0.015	0.778	0.993	0.15	0.056	2.554
	0.100	0.370	0.370	0.136	0.110	0.015	0.778	0.993	0.15	0.056	2.554
	0.321	0.382	0.382	0.101	0.122	0.012	0.495	0.692	0.65	0.038	2.914
	0.321	0.380	0.380	0.112	0.116	0.013	0.589	0.774	0.50	0.040	2.507
Rayleigh	0.100	0.361	0.361	0.127	0.135	0.017	0.534	0.973	0.20	0.056	3.306
	0.100	0.371	0.371	0.136	0.108	0.015	0.806	0.988	0.15	0.056	2.645
	0.321	0.380	0.380	0.102	0.125	0.013	0.486	0.710	0.60	0.038	2.555
	0.321	0.378	0.378	0.113	0.118	0.013	0.585	0.789	0.45	0.040	2.284
Log-logistic	0.100	0.377	0.377	0.106	0.126	0.013	0.498	0.747	0.40	0.039	2.152
	0.100	0.377	0.377	0.106	0.126	0.013	0.498	0.747	0.40	0.039	2.152
	0.321	0.383	0.383	0.102	0.120	0.012	0.511	0.695	0.75	0.038	2.999
	0.321	0.372	0.372	0.114	0.127	0.015	0.524	0.824	0.60	0.043	1.793
Log-normal	0.100	0.381	0.381	0.098	0.128	0.013	0.445	0.675	0.30	0.038	2.794
	0.100	0.372	0.372	0.120	0.120	0.015	0.603	0.868	0.20	0.045	1.880
	0.321	0.383	0.383	0.101	0.121	0.012	0.499	0.688	0.55	0.038	3.005
	0.321	0.372	0.372	0.115	0.126	0.014	0.535	0.828	0.35	0.043	1.788

For log-logistic distribution the minimal ED and MD were obtained for the parameter β = 1.5 if θ = 0.100, and if θ = 0.321 then β = 3.0 for minimal ED and β = 4.0 for minimal MD. For log-normal distribution the minimal ED and MD were reached for σ = 0.75 if θ = 0.100, and if θ = 0.321 then σ = 0.25 for minimal ED and MD.