Quantum-like behavior without quantum physics II. A quantum-like model of neural network dynamics

Abstract

In earlier work, we laid out the foundation for explaining the quantum-like behavior of neural systems in the basic kinematic case of clusters of neuron-like units. Here we extend this approach to networks and begin developing a dynamical theory for them. Our approach provides a novel mathematical foundation for neural dynamics and computation which abstracts away from lower-level biophysical details in favor of information-processing features of neural activity. The theory makes predictions concerning such pathologies as schizophrenia, dementias, and epilepsy, for which some evidence has accrued. It also suggests a model of memory retrieval mechanisms. As further proof of principle, we analyze certain energy-like eigenstates of the 13 three-neuron motif classes according to our theory and argue that their quantum-like superpositional nature has a bearing on their observed structural integrity.

Appendices

Appendix A: Some multilinear algebra

References for this appendix include [39–43].

1.1 A.1 Tensor products

We take vector spaces over a field (or modules over a ring) k which we may take to be $\mathbb {R}$, the field we will be using here. Forn of them, $V_{1}, V_{2}, {\ldots } ,V_{n}$ and another one W (they do not need to be finite dimensional here) one may have multilinear functions $f\colon V_{1} \times V_{2} \times {\ldots } \times V_{n} \rightarrow W$ meaning that f is linear in each variable separately. Linear means it preserves vector addition and scalar multiplication. One might envisage a complicated theory of such multilinear functions generalizing the theory of linear functions of a single variable. However such a theory is not necessary, since for any such collection of vector spaces $V_{1}, V_{2}, \ldots , V_{n}$ there exists a single vector space $T(V_{1}, V_{2}, {\ldots } ,V_{n})$ satisfying the following Universal Mapping Property (UMP):

There exists a multilinear map $\iota \colon V_{1} \times V_{2} \times {\ldots } \times V_{n} \rightarrow T(V_{1}, V_{2}, \ldots , V_{n})$ such that for any multilinear map $f \colon V_{1} \times V_{2} \times {\ldots } \times V_{n} \rightarrow W$ there exists a unique linear map $\tilde {f} \colon T(V_{1}, V_{2}, \ldots , V_{n}) \rightarrow W$ such that $\tilde {f}\circ \iota = f$.

It is easy to prove that such a $T(V_{1}, V_{2}, {\ldots } ,V_{n})$ must be unique up to isomorphism of vector spaces. It is called the tensor product of the vector spaces involved and written $V_{1}\otimes V_{2} \otimes \dots \otimes V_{n} = \bigotimes ^{n}_{i = 1}V_{i}$. Usually the base field or ring is appended to the tensor sign as in $\otimes _{k}$ since often algebraists have many fields and/or rings to deal with simultaneously. If the field is not in doubt, as here, it is omitted. In this way, multilinear maps are turned into linear ones and there is no need for a separate theory. One may think of the tensor product as the vector space generated by basis vectors of the form $a_{1}\otimes a_{2} \otimes \dots \otimes a_{n}, a_{i} \in V_{i}$ subject to linear additivity in each variable and the scalar multiplication property $\lambda a_{1} \otimes a_{2} \otimes \dots \otimes a_{n} = a_{1} \otimes \lambda a_{2} \otimes \dots \otimes a_{n}= {\ldots } =a_{1} \otimes a_{2} \otimes \dots \otimes \lambda a_{n}, \lambda \in k$. If $V_{i}$ has dimension $d_{i}$ then dim $(\bigotimes ^{n}_{i = 1}V_{i})= d_{1}d_{2} {\ldots } d_{n}$.

Properties of tensor products may be derived entirely through the use of the UMP stated above. For instance, suppose linear maps $f_{i}: V_{i} \rightarrow W_{i}$, $i = 1,\ldots ,n$ are given. Then $f_{1} \times {\ldots } \times f_{n}: V_{1} \times {\ldots } \times V_{n} \rightarrow W_{1} \times {\ldots } \times W_{n}$ is multilinear so that its composition with the linear $\iota$ -map of $W_{1} \times {\ldots } \times W_{n}$ into $W_{1} \otimes \dots \otimes W_{n}$ is also multilinear so that it may be lifted to produce a linear map $V_{1} \otimes \dots \otimes V_{n} \rightarrow W_{1} \otimes \dots \otimes W_{n}$ as in the diagram above. This map is denoted $\bigotimes _{i = 1}^{n} f_{i} = f_{1} \otimes \dots \otimes f_{n}$.

1.2 A.2 Exterior algebra

With notation as in the last section, let $V^{p}\!:=\overbrace {V\!\times {\ldots } \times \!V}^{p}$ for $p\geqslant 2$. Then, a multilinear function $f : V^{p} \rightarrow W$ is said to be alternating if $f(v_{1},v_{2}, \dots , v_{i},v_{i}, \ldots , v_{p}) = 0$. It follows from multilinearity that interchanging any pair of adjacent variables changes the sign of the value of f and from this that the same holds for the interchange of any pair of variables. Then it follows that the repetition of any pair of variables causes the value of f to vanish. There is a UMP for alternating maps similar to the one for general multilinear maps. The unique vector space that plays the rôle of the tensor product $\bigotimes ^{p}V$ in this case is written $\bigwedge ^{p} V$, and called the exterior product. It is generated by elements of the form $v_{1}\wedge v_{2} \wedge {\ldots } \wedge v_{p}$, $v_{i} \in V$. This element is multilinear in its arguments and alternating in the sense described above for f. Thus for instance $v \wedge v = 0$ and if $v \wedge w = 0$ then v and w generate the same one-dimensional subspace, i.e., arecolinear. (For, ifv andw were not linearly dependent they could be included in a basis forV in which case $v \wedge w$ would be a basis element of $\bigwedge ^{2}V$ which could not be the zero vector.) The map corresponding to $\iota$ in the last section sends $(v_{1}, \dots , v_{p})$ to $v_{1}\wedge v_{2} \wedge {\ldots } \wedge v_{p}$. It is not hard to show that, if the dimension ofV isn, then dim$\bigwedge ^{p} V= \binom {n}{p}$. Note that dim$\bigwedge ^{n} V = 1$ and that $\bigwedge ^{p} V = \{0\}$ if $p>n$. A useful intuition is that the exterior product $v_{1}\wedge v_{2} \wedge {\ldots } \wedge v_{p}$ is a vector representing the volume of the polytope bounded by the vectors $v_{1}, {\ldots } , v_{p}$ normal to the surface of this polytope.

These exterior products may be assembled into a unital associative algebra (i.e., an associative algebra containing a unit) having a certain universal property with respect to linear maps $f:V\rightarrow A$ into such an algebra A having the property that $f(v)^{2}= 0$ for all $v \in V$. Namely, there exists an associative unital algebra $E(V)$ for any vector space V, and a linear map $\kappa : V \rightarrow E(V)$ having the property mentioned, such that if $f:V\rightarrow A$ is any linear map into any associative unital algebra A having that property, then there exists a unique algebra morphism $\tilde {f}:E(V) \rightarrow A$ such that $\tilde {f} \circ \kappa = f$. $E(V)$ is then necessarily unique up to algebra isomorphism. One may take this algebra to be the exterior algebra of V defined by

$$ E(V):= \bigoplus\limits_{k\geqslant 0} \bigwedge^{k} V. $$

(9.28)

In case V is finite dimensional, of dimension n, say, this direct sum terminates to give

$$\begin{array}{@{}rcl@{}} E(V)& =& \bigwedge^{0} V\oplus \bigwedge^{1}V \oplus \bigwedge^{2} V \oplus \dots\oplus \bigwedge^{n} V \end{array} $$

(9.29)

$$\begin{array}{@{}rcl@{}} &=&\mathbb{R} \oplus V \oplus \bigwedge^{2} V \oplus \dots\oplus \bigwedge^{n} V. \end{array} $$

(9.30)

Here we take $\bigwedge ^{0} V = \mathbb {R}$ and $\bigwedge ^{1} V = V$. Note that dim E (V) = 2ⁿ. The algebra multiplication is given by wedging together two elements of the summands—called homogenous elements—in the order given, and extending by linearity to the whole space, with elements in $\bigwedge ^{0} V = \mathbb {R}$ just acting as scalars in the usual way. The map $\kappa : V \rightarrow E(V)$ is given by $\kappa (v) = v$ considered to lie in the summand $\bigwedge ^{1} V = V$. This algebra has many interesting properties and symmetries which were understood by H. Grassmann in the middle of the 19th century but whose published treatment of it was famously misunderstood by his contemporaries, probably because of limitations in the notations of the time. We shall rehearse a few of these properties here. First we note that for two finite dimensional vector spaces V and W the map

$$ \bigwedge^{m} V \otimes \bigwedge^{n} W \longrightarrow \bigwedge^{m +n}(V \oplus W) $$

(9.31)

given by $(v_{1}\wedge {\ldots } \wedge v_{m}) \otimes (w_{1} \wedge {\ldots } \wedge w_{n}) \mapsto v_{1}\wedge {\ldots } \wedge v_{m}\wedge w_{1} \wedge {\ldots } \wedge w_{n}$ induces an isomorphism of vector spaces

$$ \bigoplus\limits^{p}_{k = 0}(\bigwedge^{k} V\otimes \bigwedge^{p-k} W) \cong \bigwedge^{p} (V\oplus W) $$

(9.32)

from which we obtain an isomorphism of vector spaces

$$ E(V \oplus W) \cong E(V) \otimes E(W) $$

(9.33)

which is not an isomorphism of algebras when the ordinary tensor product algebra multiplication is used on the right-hand side of (9.33). There is, however, an algebra product structure on the right-hand side that does render that isomorphism above an isomorphism of algebras: it is called the graded product and it is described as follows. For homogeneous elements $a,c \in E(V)$, $b,d \in E(W)$, the graded product on the algebra $E(V) \otimes E(W)$ is determined by the definition:

$$ (a\otimes b)(c \otimes d):= (-1)^{\text{deg}(b)\text{deg}(c)} (ac \otimes bd), $$

(9.34)

where the degree deg (f) of a homogeneous element f is the power of the exterior product to which it belongs, also called the grade of f.

Let us consider the case when V is one-dimensional, with basis element e, say. Then clearly $E(V) =E(\mathbb {R}e)= \mathbb {R} \oplus V \cong \mathbb {R} \oplus \mathbb {R}e$ in our earlier notation, where, as an element in the algebra of $E(\mathbb {R}e)$, $e^{2} = e \wedge e = 0$. It is immediate that $E(\mathbb {R}e)$ is commutative as an algebra. Now, for any finite dimensional vector space V with basis $\{e_{1}, \ldots , e_{n}\}$ we have

$$ E(V ) \cong E(\mathbb{R}e_{1} \oplus \dots\oplus \mathbb{R}e_{n}) \cong E(\mathbb{R}e_{1}) \otimes \dots\otimes E(\mathbb{R}e_{n}) $$

(9.35)

as vector spaces. As noted above, this is not an isomorphism of algebras with the ordinary tensor product multiplication on the right-hand side, since this would be commutative as all of the $E(\mathbb {R}e_{i})$ are, while the left hand side is not. However, as mentioned, the right-hand side with graded product is isomorphic with the exterior product on the left-hand side. Thus, the exterior algebra may be described in terms of graded tensor products of algebras isomorphic with $E(\mathbb {R}e)$ (cf. [42]). The reader may note the similarity of such tensor products to the notion of qubit registers in the parlance of quantum computation.

If $f:V \rightarrow W$ is a linear map of vector spaces, there is a unique map of algebras $E(f): E(V) \rightarrow E(W)$ that extends $f:V \rightarrow W$. This may be proved using the UMP for exterior algebra. It is easy to see that it is given by the linear extension of the assignments $E(f)(1) = 1, E(f)(v_{1}\wedge {\ldots } \wedge v_{k}) = f(v_{1})\wedge {\ldots } \wedge f(v_{k})$.

The exterior algebra of a finite dimensional k vector space V has additional elements of structure which are not usually invoked in the physical context of the Fermi–Dirac space of a many fermion system, but will be of interest to us here. First consider the vector space maps given by $V \rightarrow V \oplus V$, $v \mapsto (v,v)$ and $V \rightarrow \{ 0\}$, $v \mapsto 0$. As in Appendix A.2 these respectively induce algebra maps

• $\psi _{V}: E(V) \rightarrow E(V\oplus V)\cong E(V) \otimes E(V)$ called the coproduct which is an algebra map given on elements $v \in V \subset E(V)$ by

  $$ \psi_{V}(v) = 1\otimes v + v \otimes 1. $$

  (9.36)

  This coproduct makes $E(V)$ a coalgebra;

• $c_{V}: E(V) \rightarrow E(\{0\})=k$, called the counit, given by the projection of $E(V)$ onto its first component.

Together with the algebra structure these maps give $E(V)$ the structure of a Hopf algebra in which the product, coproduct, unit, and counit intertwine in certain ways that need not concern us here.

1.3 A.3 The Plücker embedding

For a (real) vector space V, let $\mathbf {Gr}(p, V)$ denote the family of subspaces ofV of dimension p, the notation $\mathbf {Gr}$ being in honor of Grassmann. The special case of $p = 1$ is called the projective space of V, and is denoted by $\mathbf {P}(V)$. Exterior products may be used to obtain an explicit representation of $\mathbf {Gr}(p, V)$, namely the map

$$ \psi:\mathbf{Gr}(p, V) \longrightarrow \mathbf{P}(\bigwedge^{p} V) $$

(9.37)

given, for a p-dimensional subspace $W\subseteq V$, with basis $\{w_{1}, \ldots ,w_{p}\}$, by

$$ \psi(W) := \mathbb{R}w_{1}\wedge {\ldots} \wedge w_{p} $$

(9.38)

is well defined. For, another basis of W is related to this basis by a matrix with a non-vanishing determinant and the corresponding exterior product is the previous one, namely $w_{1}\wedge {\ldots } \wedge w_{p}$, multiplied by this determinant and so specifies the same element in $\mathbf {P}(\bigwedge ^{p} V)$. Moreover, it is not hard to see that $w\in W$ if and only if $w\wedge \psi (W) = 0$, showing that $\psi$ is one-to-one, or injective. It is called the Plücker embedding.

Intuitively this last result can be interpreted as follows: if the volume of the $(p + 1)$-dimensional polytope obtained by adding w as another side to the p-dimensional polytope with sides $w_{1}, \ldots , w_{p}$ is zero, then w must lie in the polytope, and conversely. This is easily seen when $p = 2$: if adding a third vector to the two-dimensional polytope with sides $w_{1}, w_{2}$ produces a (three-dimensional) polytope of zero volume, then w must lie in the plane determined by $w_{1}, w_{2}$, and conversely.

Appendix B: Some probabilities

The statistics of neuronal firing is a very large field but the assumption that it is a Poisson process seems to be reasonable as a general approximation. In that case, the probability of a firing at any particular time t is zero. Thus, from (8.19)

$$\begin{array}{@{}rcl@{}} &&\text{prob}(J_{ij}(t) = 0) \\ & &\ = \text{prob}((\text{v}_{ij}(t)< {\Theta}_{j})\text{ and} (n_{j} \text{ is not firing at time} \ t)) \end{array} $$

(9.39)

$$\begin{array}{@{}rcl@{}} & &\ = \text{prob}(\text{v}_{ij}(t)< {\Theta}_{j})\text{ prob}(n_{j} \text{ is not firing at time} \ t) \end{array} $$

(9.40)

$$\begin{array}{@{}rcl@{}} &&\phantom{AAaaa}\text{since these events are independent,} \\ & &\ = \text{prob}(\text{v}_{ij}(t)< {\Theta}_{j}). \end{array} $$

(9.41)

To estimate this probability we note that the full range of membrane potential values available to a typical neuron goes from $-70$ mV to 40 mV, approximately, with an approximate value of the threshold value being $-50$ mV. We therefore estimate the last named probability by taking the ratio of possible range of values of $\text {v}_{ij}(t)< {\Theta }_{j}$ to the range of all possible values for $\text {v}_{ij}(t)$ to obtain

$$ \text{prob}(J_{ij}(t) = 0)= \frac{20\text{ mV}}{110\text{ mV}} = 0.1\dot{8}. $$

(9.42)

Similarly, we find from (8.19), that

(J_ij(t) > 0) iff $\left ((\text {v}_{ij}(t)>0)\right .$ and (($\text {v}_{ij}(t)\geqslant {\Theta }_{j}$) or (n_j is firing at t)).

or

(J_ij(t) > 0) iff ((v_ij(t) > 0) or (n_j is firing at t)) since ${\Theta }_{j} = -50$ mV.

Consequently,

$$\begin{array}{@{}rcl@{}} \text{prob}(J_{ij}(t) >0) &=& \text{prob}(\text{v}_{ij}(t)>0) \end{array} $$

(9.43)

$$\begin{array}{@{}rcl@{}} &=&\frac{40\text{ mV}}{110\text{ mV}} \end{array} $$

(9.44)

$$\begin{array}{@{}rcl@{}} &=&0.3\dot{6}. \end{array} $$

(9.45)

It follows that

$$ \text{prob}(J_{ij}(t) < 0) = \frac{50\text{ mV}}{110\text{ mV}}= 0.4\dot{5}. $$

(9.46)

Also,

$$\begin{array}{@{}rcl@{}} &&\text{prob}(J_{ij}(t)J_{kl}(t) > 0) \\ && \ = \text{prob}((J_{ij}(t)>0 \text{ and} J_{kl}(t)> 0)\text{ or} \\ &&\phantom{aaaaaaaaaaa}(J_{ij}(t)<0 \text{ and} J_{kl}(t)<0)) \end{array} $$

(9.47)

$$\begin{array}{@{}rcl@{}} && \ = \text{prob}(J_{ij}(t)>0 \text{ and} J_{kl}(t)> 0)\,+\\ &&\phantom{aaaaaa} + \text{prob}(J_{ij}(t)<0 \text{ and} J_{kl}(t)<0) \end{array} $$

(9.48)

$$\begin{array}{@{}rcl@{}} &&\ = \text{prob}(J_{ij}(t)>0)\,\text{prob}(J_{kl}(t)>0) \,+\\ &&\phantom{aaaaaa} +\text{prob}(J_{ij}(t)<0)\,\text{prob}\left( (J_{kl}(t)<0)\right. \end{array} $$

(9.49)

$$\begin{array}{@{}rcl@{}} && \ = (0.3\dot{6})^{2} + (0.4\dot{5})^{2} \end{array} $$

(9.50)

$$\begin{array}{@{}rcl@{}} &&\ = 0.34 \end{array} $$

(9.51)

while

$$ \text{prob}(J_{ij}(t)J_{kl}(t)J_{mn}(t) \ne 0) = 1. $$

(9.52)

Now consider two randomly chosen two-dimensional real vectors $\mathbf {v}$ and $\mathbf {w}$. Since $\mathbf {v}.\mathbf {w}$ $= \,||\mathbf {v}||\,||\mathbf {w}|| \cos \theta _{\mathbf {v},\mathbf {w}}$, where $\theta _{\mathbf {v},\mathbf {w}}$ denotes the angle subtended by the vectors $\mathbf {v}$ and $\mathbf {w}$, we have

$$\begin{array}{@{}rcl@{}} \text{prob}(\mathbf{v}.\mathbf{w} >0) &=& \text{prob}(\cos \theta_{\mathbf{v},\mathbf{w}}>0) \end{array} $$

(9.53)

$$\begin{array}{@{}rcl@{}} & =& \text{prob}\left( \theta_{\mathbf{v},\mathbf{w}} \in (-\frac{\pi}{2}, \frac{\pi}{2})\right) \end{array} $$

(9.54)

$$\begin{array}{@{}rcl@{}} &=& \frac{\pi}{2\pi} \end{array} $$

(9.55)

$$\begin{array}{@{}rcl@{}} & =& 0.5. \end{array} $$

(9.56)