We consider a generalization of the well-known nonlinear Nicholson blowflies model with stochastic perturbations. Stability in probability of the positive equilibrium of the considered equation is studied. Two types of stability conditions: delay-dependent and delay-independent conditions are obtained, using the method of Lyapunov functionals and the method of linear matrix inequalities. The obtained results are illustrated by numerical simulations by means of some examples. The results are new, and complement the existing ones.

We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter β2 > 0, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range 0 < β2 < 4π via the variational approach due to Barashkov-Gubinelli (2018). We then prove almost sure global well-posedness and invariance of the Gibbs measure under the hyperbolic SdSG dynamics in the range 0 < β2 < 2π. Our construction of the Gibbs measure also yields almost sure global well-posedness and invariance of the Gibbs measure for the parabolic sine-Gordon model in the range 0 < β2 < 4π.

We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space–time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).

In this paper we discuss various aspects of invariant measures for nonlinear Hamiltonian partial differential equations (PDEs). In particular, we show almost-sure global existence for some Hamiltonian PDEs with initial data of the form ‘a smooth deterministic function + a rough random perturbation’ as a corollary to the Cameron–Martin theorem and known almost-sure global existence results with respect to Gaussian measures on spaces of functions.

This paper is concerned with the numerical approximations of semi-linear stochastic partial differential equations of elliptic type in multi-dimensions. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method. Numerical results demonstrate the good performance of the spectral method.

Let 𝒩* be a Hilbert inductive limit and X a Banach space. In this paper, we obtain a necessary and sufficient condition for an analytic mapping Ψ:𝒩*↦X to have a factorization of the form Ψ=T∘ℰ, where ℰ is the exponential mapping on 𝒩* and T:Γ(𝒩*)↦X is a continuous linear operator, where Γ(𝒩*) denotes the Boson Fock space over 𝒩*. To prove this result, we establish some kernel theorems for multilinear mappings defined on multifold Cartesian products of a Hilbert space and valued in a Banach space, which are of interest in their own right. We also apply the above factorization result to white noise theory and get a characterization theorem for white noise testing functionals.

White noise stimuli were used to estimate second-order kernels for complex cells in cortical area VI of the macaque monkey, and drifting grating stimuli were presented to the same sample of neurons to obtain orientation and spatial-frequency tuning curves. Using these data, we quantified how well second-order kernels predict the normalized tuning of the average response of complex cells to drifting gratings.

The estimated second-order kernel of each complex cell was transformed into an interaction function defined over all spatial and temporal lags without regard to absolute position or delay. The Fourier transform of each interaction function was then computed to obtain an interaction spectrum. For a cell that is well modeled by a second-order system, the cell’s interaction spectrum is proportional to the tuning of its average spike rate to drifting gratings. This result was used to obtain spatial-frequency and orientation tuning predictions for each cell based on its second-order kernel. From the spatial-frequency and orientation tuning curves, we computed peaks and bandwidths, and an index for directional selectivity.

We found that the predictions derived from second-order kernels provide an accurate description of the change in the average spike rate of complex cells to single drifting sine–wave gratings. These findings are consistent with a model for complex cells that has a quadratic spectral energy operator at its core but are inconsistent with a spectral amplitude model.

We have used Sutter's (1987) spatiotemporal m-sequence method to map the receptive fields of neurons in the visual system of the cat. The stimulus consisted of a grid of 16 X 16 square regions, each of which was modulated in time by a pseudorandom binary signal, known as an m-sequence. Several strategies for displaying the m-sequence stimulus are presented. The results of the method are illustrated with two examples. For both geniculate neurons and cortical simple cells, the measurement of first-order response properties with the m-sequence method provided a detailed characterization of classical receptive-field structures. First, we measured a spatiotemporal map of both the center and surround of a Y-cell in the lateral geniculate nucleus (LGN). The time courses of the center responses was biphasic: OFF at short latencies, ON at longer latencies. The surround was also biphasic—ON then OFF—but somewhat slower. Second, we mapped the response properties of an area 17 directional simple cell. The response dynamics of the ON and OFF subregions varied considerably; the time to peak ranged over more than a factor of two. This spatiotemporal inseparability is related to the cell's directional selectivity (Reid et al., 1987, 1991; McLean & Palmer, 1989; McLean et al., 1994). The detail with which the time course of response can be measured at many different positions is one of the strengths of the m-sequence method.

The temporal dynamics of the response of neurons in the outer retina were investigated by intracellular recording from cones, bipolar, and horizontal cells in the intact, light-adapted retina of the tiger salamander (Ambystoma tigrinum), with special emphasis on comparing the two major classes of bipolars cells, the ON depolarizing bipolars (Bd) and the OFF hyperpolarizing bipolars (Bh). Transfer functions were computed from impulse responses evoked by a brief light flash on a steady background of 20 cd/m2. Phase delays ranged from about 89 ms for cones to 170 ms for Bd cells, yielding delays relative to that of cones of about 49 ms for Bh cells and 81 ms for Bd cells. The difference between Bd and Bh cells, which may be due to a delay introduced by the second messenger G-protein pathway unique to Bd cells, was further quantified by latency measurements and responses to white noise. The amplitude transfer functions of the outer retinal neurons varied with light adaptation in qualitative agreement with results for other vertebrates and human vision. The transfer functions at 20 cd/m2 were predominantly low pass with 10-fold attenuation at about 13, 14, 9.1, and 7.7 Hz for cones, horizontal, Bh, and Bd cells, respectively. The transfer function from the cone voltage to the bipolar voltage response, as computed from the above measurements, was low pass and approximated by a cascade of three low pass RC filters (“leaky integrators”). These results for cone→bipolar transmission are surprisingly similar to recent results for rod→bipolar transmission in salamander slice preparations. These and other findings suggest that the rate of vesicle replenishment rather than the rate of release may be a common factor shaping synaptic signal transmission from rods and cones to bipolar cells.

Sample path large deviationsfor the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero arepresented. The noise is a complex additive Gaussian noise. It iswhite in time and colored in space. The solutions may be global orblow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologiesanalogue to projective limit topologies. In this setting, thesupport of the law of the solution is also characterized. As aconsequence, results on the law of the blow-up time andasymptotics when the noise converges to zero are obtained. Anapplication to the transmission of solitary waves in fiber opticsis also given.