## 2 Representing Electrochemical Uncertainty in Action Potentials by Quantum-Like Superposition  
The main idea behind our model [23] connecting electrochemical processes in neural networks of the brain with the quantum-like probabilistic structure of decision-making by humans is that neurons' states can be expressed by normalized vectors of complex Hilbert space. This formalization reflects the basic feature of quantum-like information processing by neurons:  
_A single neuron has to 'work' with probabilistic superpositions of states carrying cognitive information._  
Another crucial point is that a neural cell cannot be considered as an isolated bio-physical system. Its state of superposition is coupled with superposition states of neurons in a neural network performing some task. A neuron is treated as an open system, in which its state is affected by information flows from the electrochemical environment composed of signals (electrical currents, electromagnetic field, neurotransmitters, etc.) generated by other neurons. We recall that the classical information approach to modeling of brain functioning explores discrete states of a neuron corresponding to various neural codes [37], e.g., \(0/1\), _quiescent/firing neural code._ We stress the role of states of uncertainty, e.g., _neither quiescent nor firing_, in information processing. Such states can be represented as superpositions. Our aim is to lift processing of such states to the quantum-like statistics of outputs of a psychological function \(F\). We propose to describe uncertainty in generation of an action potential by a neuron, say \(\mathcal{N}\), by using states' superposition. Consider two dimensional complex Hilbert space \(H\) (qubit space). At a concrete instant of time, a neuron's state can be mathematically described by a superposition of two states, _quiescent and firing_, labeled by \(|0\rangle,|1\rangle\) :  
\[|\psi\rangle=c_{0}|0\rangle+c_{1}|1\rangle,\ |c_{0}|^{2}+|c_{1}|^{2}=1. \tag{1}\]  
It is assumed that these states are orthogonal and normalized, i.e., \(\langle 0|1\rangle=0\) and \(\langle\alpha|\alpha\rangle=1,\alpha=0,1\). Here, the coordinates \(c_{0}\) and \(c_{1}\), with respect to the quiescent-firing basis, are complex amplitudes representing potentialities for a neuron to be quiescent or firing. Probabilistically, these potentialities are expressed by _the Born rule of quantum theory._  
\[p_{\alpha}=|c_{\alpha}|^{2}=|\langle\psi|\alpha\rangle|^{2}. \tag{2}\]  
We proceed with the frequency approach to probability. In the conventional classical model of probability ([PERSON], 1933 [38]), this approach, i.e., coupling of probability with the frequency of occurrences, is based on the law of large numbers; see also [PERSON] frequency theory [39]. Let \(|\psi\rangle\) be a steady state, with respect to the dynamics realizing a psychological function (see Section 4). Consider a single neuron \(\mathcal{N}\) and a sufficiently long time interval \(T=M\Delta,M>>1\), and find the frequency \(v_{M}(1)=n_{M}(1)/M\), where \(n_{M}(1)\) is the number of \(\Delta\)-intervals such that neuron \(\mathcal{N}\) produces a spike. Then, \(p_{1}\approx v_{M}(1)\). We repeat that steady states play an exceptional role in our model as decision states (see Sections 6.2, 7 and works in Reference [2; 15; 16; 23]). An observer can count spikes. But _who is the observer?_ In our model, the brain (more concretely, each psychological function) is a system that is able to perform _self-observations_: detection of information states of neurons and neural networks--but not electrochemical states, cf. with ontic-epistemic description. Consider now a group of neurons \(G\) (say \(m\) neurons) connected into a neural network. Quantum information state space of \(G\) is given by tensor product of state spaces for individual neurons.