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any externally given time series x ( t ) for the input, the adaption rules (4) will lead to a distribution of the out- put firing rates y ( t ) as close as possible, given the spec- ification (1) for the transfer function, to an exponential with given mean µ . 

Attractor relics. – In deriving the stochastic gradient rules (4) it has been assumed that the input x ( t ) is sta- tistically independent of the output y ( t ) (but not vice versa). This is not any more the case when a set of poly- homeostatically adapting neurons are mutually intercon- nected, forming a recurrent network, Here we will show that networks based on polyhomeostatic principles will generically show spontaneous and continuously ongoing activity. 

In a first step we analyze systematically the smallest possible network, viz the single-site loop, obtained by feeding the output back to the input, see Fig. 1, a setup which has been termed in a different context ‘autapse’ [8]. We use the balanced substitution x → y − 1 / 2 in Eqs. (1) and (4), the complete set of evolution rules for the dynamical variables y ( t ) , a ( t ) and b ( t ) is then 

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with 

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Note, that Δ ˜ b ( t ) in (6) depends on y ( t + 1), and not on y ( t ), as one can easily verify when going through the derivation of the rules (4) for the intrinsic plasticity. The evolution equations (5) are invariant under y ↔ (1 − y ), b ↔ $^{(}$− b ), a ↔ a and λ ↔ $^{(}$− λ ), the later corresponding to the interchange of µ $^{(1}$− µ ). We first consider the quasistatic limit ϵ$_{a}$ ϵ$_{b}$ , viz 

≪ a ( t ) ≃ a is approximatively constant. The fixpoint ( y $^{∗}$, b$^{∗}$) in the ( y, b ) plane is then determined by 

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A straightforward linear stability analysis shows, that the fixpoint ( y $^{∗}$, b$^{∗}$) remains stable for small gains a and becomes unstable for large gains, see Fig. 1. We now go beyond the quasistatic approximation and consider a small but finite polyhomeostatic adaption rate ϵ$_{a}$ for the gain. Starting with a small gain a we see, compare Eq. (5), that the gain necessarily grows until it hits the boundary towards instability; for small Δ ˜ b the growths of the gain is a ( t ) ∼ √$_{t}$$_{.}$ In other words, a finite adaption rate ϵ$_{a}$ for the gain 

turns the fixpoint attractor ( y $^{∗}$, b$^{∗}$) into an attractor relic and the resulting dynamics becomes non trivial and autonomously self-sustained. This route towards au- tonomous neural dynamics is actually an instance of a more general principle. Starting with an attractor dy- namical system, viz with a system governed by a finite number of stable fixpoint, one can quite generally turn these fixpoints into unstable attractor ruins by coupling locally to slow degree of freedoms [10, 11], in our case the slow local variables are a ( t ) and b ( t ). Interestingly, there are no saddlepoints present in attractor relic networks in general, and in (5) in particular, and hence no unstable heteroclines, as in a proposed alternative route to tran- sient state dynamics via heteroclinic switching. [12, 13] 

In Fig. 2 the time evolution is illustrated for λ = 3 . 017, which corresponds to µ = 0 . 28, see Eq. (2), and ϵ$_{a}$ = ϵ$_{b}$ = ϵ = 0 . 01. The system remains in the quasistation- ary initial regime until the gain a surpasses a certain threshold. The initial quasistationary fixpoint becomes therefore unstable via the same mechanism discussed an- alytically above, see Eq. (7), for the regime ϵ$_{a}$ → 0, com- pare Fig. 1. The output activity y ( t ) oscillates fast be- tween two transient fixpoints of y = g ( a [ y − 1 / 2] + b ), having a high and a low value respectively. This spiking behavior of the neural activity is driven by spontaneous oscillations in the threshold − b ( t ) /a ( t ), shifting the in- tersection of g ( a [ y − 1 / 2] + b ) with y forth and back. 

