Explicit logic circuits predict local properties of the neocortex's physiology and anatomy

Abstract

Background: Two previous articles proposed an explicit model of how the brain processes information by its organization of synaptic connections. The family of logic circuits was shown to generate neural correlates of complex psychophysical phenomena in different sensory systems.

Methodology/principal findings: Here it is shown that the most cost-effective architectures for these networks produce correlates of electrophysiological brain phenomena and predict major aspects of the anatomical structure and physiological organization of the neocortex. The logic circuits are markedly efficient in several respects and provide the foundation for all of the brain's combinational processing of information.

Conclusions/significance: At the local level, these networks account for much of the physical structure of the neocortex as well its organization of synaptic connections. Electronic implementations of the logic circuits may be more efficient than current electronic logic arrays in generating both Boolean and fuzzy logic.

Figure 1. Recursive AND NOT Conjunctions.

A Recursive AND NOT Conjunction (RANC) is a general logic circuit that produces conjunctions, with negations, and is defined recursively in terms of AND NOT gates. An n-RANC produces conjunctions of n propositions. A Complete n-RANC produces all conjunctions corresponding to the 2ⁿ possible combinations of truth values of n propositions. Examples of Complete n-RANCs are shown here for n = 1–4. The label on each neuron represents its response. The maximum and minimum possible responses 1 and 0 can stand for the logical values true and false, making the network outputs logical functions of the inputs. Arrows indicate excitatory input; blocks indicate inhibitory input. Spontaneously active neurons are square. To illustrate example inputs and outputs, active neurons are colored. Inactive inhibitory cells are shaded. The line graphs below each circuit diagram illustrate the RANC interval measure property and the Boolean and fuzzy logic of the example inputs and outputs.

Figure 2. A Recursive AND NOT Conjunction with several inputs.

The circuit diagram shows nearly half of a Complete 6-RANC. It illustrates several general properties of BC n-RANCs, including properties that hold for n>4. The second and third columns have shared inhibitory cells. The example inputs produce active regions surrounded by neurons that are inhibited below the resting level. The example inputs also show that information is processed in small columns and is transformed in each layer while passing through the columns in series and in parallel. The diagram illustrates the modular architecture that is determined by the recursive definition of a BC n-RANC. The line graph shows the RANC interval measure property and the fuzzy logic of the approximate outputs. For example, the output number 22,  = 0.20, represents the truth value of the conjunction “X₂, X₃, and X₅ are true, and X₁, X₄, and X₆ are false.”

Figure 3. Non-hexagagonal columns.

A fourth of a BC n-RANC's outputs are not produced by hexagonal columns. This includes the n-RANCs for the outputs labeled 2ⁿ⁻¹−1 and 2ⁿ⁻¹. These n-RANCs are shown in A, B, C for n = 4, 5, 6, respectively. To transform each single n-RANC into an (n+1)-RANC, n excitatory cells and one inhibitory cell are added as shown in blue. Other non-hexagonal n-RANCs are formed from these by the recursive definition of a BC n-RANC. The input cells that are not shown and the cells shown in orange are not part of the networks because they are either inputs to the BC n-RANC or part of other n-RANCs within the BC n-RANC.

Figure 4. Modular architecture with repeated parts.

A RANC hexagonal column begins with unconnected cells in A. Inhibitory connections in B form identical three-cell parts in different layers. Excitatory connections in C combine these parts into a three-layer module. Additional excitatory connections in D join three identical modules to form a column. When several columns share inhibitory cells, as illustrated in Fig. 2, modular construction is nearly as simple. Inhibitory connections in E form identical parts in different layers, and excitatory connections in F combine these parts into a three-layer module. Additional excitatory connections in G join three of these modules to form three columns.

Figure 5. Number of cells required.

The directed graphs show that hexagonal columns nearly minimize the number of cells used to construct Complete n-RANCs. Two nodes are linked if they can form a conjunction according to equation 1 or 2 of Table 2. Along each edge, the label representing the response is shown next to node A, and is next to B. The edge labels in bold face show the conjunctions as they are implemented in Fig. 1C, D. The graph in A shows all the ways six single 3-RANCs can be formed, and B shows how six single 3-RANCs can be formed with one less cell. The graph in C shows how 4-RANCs can be formed.

Figure 6. Connection length and cell packing.

The connectivity of RANCs suggests an arrangement of cells that nearly optimizes total connection length and cellular packing density. Packing efficiency is illustrated with cell bodies contained in virtual spheres whose diameters are determined by the physiological constraints of necessary separations between cell bodies. Connection length is minimized when two connected cells are in adjacent spheres. The three layers of a typical hexagonal column are shown in A, with the front view of the column in B. Five layers of cells that produce outputs 1–6 in a Complete 7-RANC are shown in C, with the side and front views in D and E. The lower three layers of three columns with shared cells are shown in F and G.

Figure 7. Inhibitory cells in BC n-RANCs.

The graph shows the BC n-RANC's percentage of inhibitory cells as a function of the number of inputs n. Input cells are not counted as part of the network. In the cortex, 20 to 25% of the neurons are inhibitory. This agrees well with BC n-RANCs, which have between 20 and 25% inhibitory neurons for all n≥7. The proportion is about 23.6% for n = 7. As n increases, the proportion decreases asymptotically to about 20.4%, shown in green in the graph.