Table 2. The best optimal theoretical fractional volumes and related parameters in the cortex for a particular case of “wire minimization” principle associated with wire volume minimization (γ ₁ = 0).

The optimal fractions correspond either to minimal Euclidean distance (ED) or minimal 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	r
Exponential	0.100	0.388	0.330	0.068	0.201	0.014	0.157	0.528	0.94	0.091	6.981
	0.100	0.403	0.316	0.067	0.200	0.013	0.157	0.528	0.91	0.094	6.942
	0.321	0.423	0.371	0.111	0.085	0.009	0.935	0.709	0.96	0.045	6.485
	0.321	0.397	0.397	0.112	0.085	0.010	0.936	0.710	1.00	0.058	6.360
Gamma (n = 1)	0.100	0.376	0.316	0.081	0.210	0.017	0.175	0.684	0.93	0.104	7.842
	0.100	0.442	0.259	0.078	0.206	0.016	0.175	0.682	0.80	0.134	7.427
	0.321	0.413	0.357	0.119	0.099	0.012	0.793	0.805	0.95	0.030	3.991
	0.321	0.396	0.374	0.119	0.099	0.012	0.794	0.806	0.98	0.037	3.892
Gamma (n = 2)	0.100	0.366	0.310	0.086	0.219	0.019	0.175	0.753	0.93	0.115	9.078
	0.100	0.463	0.230	0.080	0.210	0.017	0.174	0.750	0.74	0.163	8.173
	0.321	0.406	0.352	0.121	0.108	0.013	0.715	0.846	0.95	0.026	2.681
	0.321	0.395	0.363	0.121	0.108	0.013	0.715	0.846	0.97	0.030	2.640
Rayleigh	0.100	0.368	0.305	0.082	0.226	0.019	0.159	0.734	0.92	0.124	9.623
	0.100	0.471	0.221	0.076	0.216	0.016	0.158	0.731	0.72	0.177	8.701
	0.321	0.405	0.352	0.117	0.113	0.013	0.642	0.822	0.95	0.020	2.271
	0.321	0.394	0.362	0.117	0.113	0.013	0.642	0.822	0.97	0.025	2.236
Log-logistic	0.100	0.404	0.350	0.097	0.136	0.013	0.404	0.683	0.95	0.015	2.378
	0.100	0.399	0.356	0.097	0.136	0.013	0.404	0.684	0.96	0.017	2.351
	0.321	0.399	0.348	0.124	0.114	0.014	0.671	0.891	0.95	0.027	1.803
	0.321	0.403	0.334	0.123	0.124	0.015	0.584	0.910	0.93	0.031	1.448
Log-normal	0.100	0.418	0.367	0.072	0.133	0.010	0.311	0.472	0.96	0.034	5.734
	0.100	0.401	0.356	0.073	0.158	0.012	0.244	0.513	0.96	0.045	4.801
	0.321	0.411	0.354	0.106	0.116	0.012	0.561	0.731	0.95	0.010	2.897
	0.321	0.402	0.341	0.110	0.133	0.015	0.475	0.804	0.94	0.021	1.405

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 = 3.5 for minimal ED and β = 4.5 for minimal MD. For log-normal distribution the minimal ED was reached for σ = 0.7 and minimal MD for σ = 0.55 if θ = 0.100, and if θ = 0.321 then σ = 0.3 for minimal ED and σ = 0.2 for minimal MD.