Explicit logic circuits discriminate neural states

Abstract

The magnitude and apparent complexity of the brain's connectivity have left explicit networks largely unexplored. As a result, the relationship between the organization of synaptic connections and how the brain processes information is poorly understood. A recently proposed retinal network that produces neural correlates of color vision is refined and extended here to a family of general logic circuits. For any combination of high and low activity in any set of neurons, one of the logic circuits can receive input from the neurons and activate a single output neuron whenever the input neurons have the given activity state. The strength of the output neuron's response is a measure of the difference between the smallest of the high inputs and the largest of the low inputs. The networks generate correlates of known psychophysical phenomena. These results follow directly from the most cost-effective architectures for specific logic circuits and the minimal cellular capabilities of excitation and inhibition. The networks function dynamically, making their operation consistent with the speed of most brain functions. The networks show that well-known psychophysical phenomena do not require extraordinarily complex brain structures, and that a single network architecture can produce apparently disparate phenomena in different sensory systems.

Figure 1. Neurons as functionally complete logic gates.

The circuit diagrams show that neurons with excitatory and inhibitory inputs and neurons that have continuously high outputs form a functionally complete set, meaning any logic circuit can be constructed with them. 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. The diagrams show logic gates for (A) X AND NOT Y, (B) X AND Y, and (C, D) NOT X. 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.

Figure 2. Recursive AND NOT Conjunctions.

An n-RANC is a general logic circuit that 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. A single n-RANC produces one of the possible conjunctions. In C, the single 3-RANC that produces output number 3, , is indicated by thick lines. In D, the output number 14, , represents the truth value of the conjunction “X₂, X₃, and X₄ are high, and X₁ is not high.” The other 15 conjunctions are false, and the corresponding RANC outputs are 0.

Figure 3. Fuzzy logic of a complete 4-RANC.

The figure in A shows the approximate computations of a complete 4-RANC when one of the inputs has an intermediate value between 0 and 1. The graph in B illustrates the RANC interval measure property: The output intensities (approximately 0.7 and 0.3) are measures of the subintervals ([0, 0.7], and [0.7, 1]) of [0, 1] formed by the input intensities. The combination of output cells that respond uniquely identifies the ordering of the input intensities (0 = X₃<X₄<X₁ = X₂ = 1). The response represents the fuzzy truth value of the conjunction “X₁ and X₂ are high and X₃ and X₄ are not high.”

Figure 4. Relative Absorption Model responses to a greenish-yellow photostimulus.

A greenish-yellow photostimulus moderately represses the L cone response and strongly represses the M cone so that M represents the fuzzy truth value of the conjunction “S and L are high and M is not high.” Since this state of cone responses is the condition for the perception of green, also represents the fuzzy truth value of the proposition “The photostimulus is green.” That is, is the correlate of the perceived strength of the green component of the photostimulus.